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CONTENTS OF VOL. XX. 
(SIXTH SERIES). 


NUMBER CXV,.—JULY 1910. 


Mr. John Satterly on the Amount of Radium Emanation in 
the Lower Regions of the Atmosphere and its Variation 
math the Weather. (Plates I. & IL.) .. 5.4... .seees -- 

Mr. A. L. Fletcher on the Radioactivity of the Rocks of the 
ese EMI TMP E AAT 20 cae al Sch ap sib mins # w'siin am balsa Beals 

Messrs. C. Barus and M. Barus on the Interference of the 
Reflected-Diffracted and the Diffracted-Reflected Rays of 
a Plane Transparent Grating, and on an Interferometer . 

Mr. A. Whitwell on the Lengths of the Focal Lines of 
eremraier  Werse 69's). pes) wae eid a aleyd a 3.06 s Sores Ka D aim aie 

Mr. E. Howard Smart on a Formula for the Spherical Aber- 
ration in a Lens-System correct to the Fourth Power of 
SMES CN te al Sh Biel oe BEd GS Gf ide Ste nmr wien acecei dy Se 

Messrs. G. Stead and H. Donaldson on the Problem of Uni- 
form Rotation treated on the Principle of Relativity 

Prof. A. H. Gibson on a Rational Formula for the Discharge 
Beers Droad-crested Weir 2. 2.06.06. 6 bac eke nena siee 

Prof. W. E. Story on Partial Pressures in Liquid Mixtures. . 

Prof. Harold A. Wilson on the Statistical Theory of Heat 
MOE ets ee ARS ie de spy wind, Ea hhd ate, Sa ds 

Prof. J. Joly on the Amount of Thorium in Sedimentary 
Rocks.—I. Calcareous and Dolomitic Rocks ............ 

Dr. W. H. Eccles on an Oscillation Detector actuated solely 
by Resistance-Temperature Variations .........+...... 

Dr. G. Bakker on the Theory of Surface Forces.—V. Thermo- 
dynamics of the Capillary Layer between the Homogeneous 
haces of the Liquid and the Vapour... .!- 2-6: . 5.4.0. 

Dr. J. W. Nicholson on the Bending of Electric Waves round 
enn CMe, Bal tbe gouhelana ian aan dm Th ols 


Profs. O. W. Richardson and H. L. Cooke on the Heat 


developed during the Absorption of Electrous by Platinum. 
Re ie stole dala cf Mean Sk ESN pialia he Gece dee 0 8 
Dr. 8S. W. J. Smith on the Limitations of the Weston Cell as 
a Standard of Electromotive Force ..............006. 
Dr. R. D. Kleeman on the Shape of the Atom ............ 
Sir J. J. Thomson on the Theory of Radiation ............ 
Notices respecting New. Books :— 
C. Tissot’s Les Oscillations électriques, and BE. J ouguel s 
Théorie des Moteurs Thermiques .....+.+.+.s005. 


Page 


82 
92 


95 
97 


121 


157 


173 
206 


229 
238 


247 


lv CONTENTS OF VOL. XX.—SIXTH SERIES. 


Intelligence and Miscellaneous Articles :— 
Reply to Mr. W. J. Harrison, by Dr. R. A. Houstoun . 247 
On the Homogeneity of the y Rays of Radium, by 
ie. WD. Kleeman ........ ced ines Oe 2:48 


NUMBER CXYI.—AUGUST. 


Mr. William Sutherland on Molecular and Electronic 
RoMetieia SOGOU RY 5.5 Ls. 0s we Re ee se 249 
Prof, A. P. Chattock on the Forces at the Surface of a 
Needie-Pomt discharging in Air... 39.45.54 (sae pee 266 
Prof. A. P. Chattock and Mr. A. M. Tyndall on the Ionizing 
Processes at a Point discharging in Air. (Plate LV.) .... 277 
Prof. Max Mason on the Flow of Energy’ inan Interference Field 290 
Dr. W. Miller on a Constant Pressure Gas Thermometer .. 296 
Messrs. Horace Lamb and Gilbert Cook: A Hydrodynamical 
Illustration of the Theory of the Transmission of Aerial 
and Hlectrical Waves by a Grating’ .. 2... 0... sa gene 303 
Dr. W. E. Sumpner and Mr. W. C. 8. Phillips on a Galvano- 
meter for Alternate Current Circults’..............ss00s 309 
Mr. R. T. Beatty on the Production of Cathode Particles by 
Homogeneous Réntgen Radiations, and their Absorption 


by Hy ‘drogen and Air.” ‘@Plate Viv + poeho50):. 0 eee 320 
Mr. Otto Stuhlmann on a Difference in the Photoelectric 
Effect caused by Incident and Emergent Light ........ 301 
Mr. Frederick Soddy on the Relation between Uranium and 
Badium.—V.. >... 3k os Uk we bo epee ounce a oe 340 
Mr. Frederick Soddy on the Rays and Product of Ura- 
pe ATT tk ee wane erase) ves ire 342 
Mr. Frederick Soddy and Miss Ruth Pirret on the Ratio 
between Uranium and Radium in Minerals ............ 345 


Prof. Sir J. Larmor on the Statistical Theory of Radiation .. 350 
Prof. J. Joly on the Amount of Thorium in Sedimentary 
Rocks.—II. Arenaceous and Argillaceous Rocks ........ 353 
M. C. Cheneveau on the Magnetic Balance of MM. P. Curie 
and C. Cheneveau ; with an Appendix by A. C. Jolley... 387 
Prof. Charles G. Barkla on Typical Cases of Ionization by : 
EMT et cero se eee an, Stee asa eer 370 
Prof. J. H. Jeans on the Motion of a Particle about a Doublet. 380 
Notices respecting New Books :-— 
Bulletin of the Bureau of Standards, Vol.6 .......... 382 
Inteliigence 
On the Homogeneity of the y Rays of Radium, by 


Mr. Soddy - 54. .ooe oes hee: a 383 
On the Electrostatic Effect of a Changing Magnetic 
Meld, "by Prot. J: BY Whitehead 0.07 72.2, eae 384 


On the Laws regarding the Direction of Thermo-electric 
Currents enunciated by M. Thomas, by Dr. C. H. Lees. 384 


CONTENTS OF VOL. XX.—SIXTH SERIES. 


NUMBER CXVII.—SEPTEMBER. 


Prof. W. H. Brage on the Consequences of the Corpuscular 
Hypothesis of the y and X Rays, and the Range of 
Oo LE OR ESS OR BaP SOE See ere eee ee ee 

Prof. Silvanus P. Thompson on Hysteresis Loops and Lis- 
sajous’ Figures, and on the Energy wasted in a Hysteresis 
Beare te ete VA esis dy Sat) bal ape l Waxave a! ab aaele det 

Mr. J. Prescott on the Precise Effect of Radial Forces in 
opposing the Distortion of an Elastic Sphere .......... 


Mr A: K.-H. Love: Note on the preceding Paper ...2....1 : 


Dr. R. D. Kleeman on the Shape of the Molecule ........ 
Lord Rayleigh on the Finite Vibrations of a System about 


neomucuration Of Kquilibriam ij... 0.24 gee eyed le « 
Prof. Edwin H. Barton and Mr. T. F. Ebblewhite: Vibration 


Curves of Violin Bridge and Strings. (Plates VII.-IX.) . - 


Mr. J. J. Lonsdale on the Ionization produced by the 
eee rte OE MEST OURY ps co's lea lignes lac ake aphin eer aed w aca 
Prot. W. J. Lewis: Wiltshireite: a New Mineral ........ 
Dr. C. Chree: Discussion of Results obtained at Kew Ob- 
servatory with Elster and Geitel Electrical Dissipation 
mepetains prom O07 toskG09. |. te 5 wi. een a ae Mat wwe 
Mr. W. J. Harrison on the Stability of Superposed Streams 
eee oils Waiuiany \ a Fe bi See) Jule abel ath eden Wie ed 
Dr. Wm. C. McC. Lewis on the Nature of the Transition 
Iuayer between Two Adjacent Phases ........5.00544: 
Prof. Alfred W. Porter on the Lagging ‘of Pipes and Wires: 
with an Addendum in conjunction with Mr. E. R. Martin . 
Mr. D. Tyrer on the Relations between the Physical Pro- 
perties of Liquids at the Boiling-Point ................ 
Dr. W. H. Eccles on the Energy Relations of Certain De- 
tectors used in Wireless Telegraphy. (Plate X.) ...... 
Mr. G. W. de Tunzelmann on the Mechanical Pressure of 
Radiation effective on the smallest as well as larger 


Rene mou aieken Wybeya toe dake ede Wig 2 a aa ape elds 538 
Notices respecting New Books :— 
Sir William Thomson’s (Baron Kelvin) Mathematical 
anal PlnyeiGal Papersa ii Viol. BV a tx tiga So ne Shiite « 540 » 
Mr. H. Crabtree’s Elementary Treatment of the Theory 
of Spinning Tops and Gyroscope Motion .......... 542 
Dr. E. Jahnke and F. Emde’s Funktionentafeln mit 
Ponmeliniaind J@ueven:) \uie.2 uctieais a a2 40e hg WEA 542 
Proceedings of the Geological Society .........6.222 6 -00-. 543 
Intelligence and Miscellaneous Articles :— 
On the Motion of an Electrified Particle near an 
Electrical Doublet, by Sir J. J. Thomson .......... 544 


ae 


v1 CONTENTS OF VOL, XX.—SIXTH SERIES. 


NUMBER CXVIII.—OCTOBER. 
Page 
Prot. O. W. Richardson and Mr. E. R. Hulbirt on the Specific 
Charge of the Ions emitted by Hot Bodies.—II. 
Prof. W. M. Thornton: The Eye as an Electrical Organ .. 560 
Dr. H. Stanley Allen on the Photoelectric Fatigue of Metals. 564 
Mr. A. E. Garrett on the Positive Electrification due to 


Heating Aluminium Phosphate .o0f.02..0 0...) 573 
Dr. Alexander Russell on the Convection of Heat from a 
ody cooled by a Stream of Fluid .)...0¢. 4. Ae 591 
Dr. J. W. Nicholson on the Accelerated Motien of an Electri- 
MET) BOATO. ii. sg ee nie A Ee Oe 610 
Prof. E. Goldstein on Threefold Emission-Spectra of Solid 
Aromatic Compounds 2... 20.5.0. sehen ss Sr 619 
Mr. H. Bateman on the Relation between Electromagnetism 
hg Geometry |... . oslo ss sg & oe eee ee ee 623 
J. EH. Mills on Molecuiar Attraction ......../. [2 629 
De S. R. Milner on the Series Spectrum of Mercury ...... 636 
Prof. J. H. Jeans on the Analysis of the Radiation from 
Electron Orbits \.. is sc Gaede wa) 2 642 
Mr. G. H. Berry on the Pianoforte Sounding-Board. (Plate 
| 1) Pe re 652 


Mr. W. Sutherland on the Mechanical Vibration of Atoms . 667 
Prof. E. Taylor Jonesand Mr. D. E. Roberts on Musical Are 
Oscillations in Coupled Circuits. (Plate XIII.) ........ 660 - 
Mr. Robert E. Baynes: Note on Mr. Bateman’s Paper on 
Earthquake-Waves ...70%5. 000004... .1 4. 664 
Dr. R. D. Kleeman on the Equation of Continuity of the 
Liquid and Gaseous States of Matter .............-.. 665 
Prof. H. 8. Carslaw on the Scattering of Waves by a Cone.. 690 
Dr. Hans Geiger and Prof. E. Rutherford on the Number of 
a Particles emitted by Uranium and Thorium and by 
Mranium Minerals: i .eac. ve PP). ott ee 691 
Prof. E. Rutherford and Dr. H. Geiger on the Probability 
Variations in the Distribution of a Particles: with a Note 


by H. Bateman ..4 6 J0c2 0 ge tee ade 42 2s 698 
Prof. R. W. Wood ona New Radiant Emission from the Spark. 

(Plate XLV): 2. Le ce ae he Bare TE  .| 707 
Prof. R. W. Wood: Some Experiments on Refraction by 

non-homogeneous Media. (Plate XIV. fig. 10.) ........ 712 
Dr. G. W. C. Kaye on a Method of Counting the Rulings of 

a Diffraction Grating: (Plate XV.) ............ 2 714 
Dr. G. W. C. Kaye on the Expansion and Thermal Hy steteets 

or Mused Biliea....decseceeeeee sss...) nnn 718 
Sir G. Greenhill on Pendulum Motion and Spherical THB 

moments 0. RL, cere he. OS. > {20 


Prof. E. G. Coker on the Optical Determination of Stress .. 740 


Sir J. J. Thomson on Rays of Positive Electricity ........ 752 


Prof. A. Trowbridge on a Vacuum Spectrometer. (Plate XVI.) 768 
Prof. R. W. Wood on the Echelette Grating for the Infra- 
Red. (Plate XVII.) ......0:..4.5 5... 29ers 779 


CONTENTS OF VOL. XX.—SIXTH SERIES. Vu 


Page 

Mr. John Satterly : Some experiments on the Absorption of é 

Radium nea bye Coconuts Charcoal oi. jas cen. 718 
Mr. Andrew Stephenson on Displacements in the Spectrum 

Given GO Ieressimer Wen ears wl Rte hefoie Wuiia. yaNegt ie «chet S46 os 788 

Proceedings of the Geological Society ..........406+.55- 790 

NUMBER CXIX.—NOVEMBER. 

Mr. F. B. Young on the Critical Phenomena of Ether .... 793 
Dr. J. W. Nicholson on the Accelerated Motion of a Dielectric 

Be i's eg eae nciey ek NR aT ep ia Yo hb apeg Wa lene oyeyaigk Male « 828 
Prot. Harold A. Wilson on the Electron Theory of the Optical 

meapornes or Metals 2005 p25. foie tie dane bnesdiy me wetetite.s 839 


Mr. A.Stephenson on the Intensity of Periodic Fields of Force 844 
Dr. Alois F. Kovarik on Absorption and Reflexion of the 

foe DE RTECS De VAL LOT oc carta aee, ahnyd ob ebea ns Sela wiahe sala o $ 849 
Dr. Alois F. Kovarik and Mr. W. Wilson on the Reflexion 

of Homogeneous /3-Particles of Different Velocities...... 866 
Messrs. J. A. Gray and W. Wilson on the Heterogeneity of 

the 8 Rays from a Thick Layer of Radium E .......... 870 
Drs. Sidney Russ and Walter Makower on the Deflexion 

by an Electrostatic Field of Radium B on Recoil from 

Hier espa OS Nae 0 sah he Sora, S595 Sas wks ere US Cash dhreumnen ey teeth se hie 875 
Dr. W. Makower and Mr. E. J. Evans on the Deflexion by 

a Magnetic Field of Radium B on recoil from Radium A. 


Pleyel Twi ene ec A ll ke. Gy Seb od. 882 
Profs. A. Trowbridge and R. W. Wood on Groove-Form and 
Energy Distribution cf Diffraction Gratings............ 886 
Profs. A. Trowbridge and R. W. Wood on Infra-Red In- 
vestigations with the Evhelette Grating................ 898 
Dr. R. D. Kleeman on Molecular Attraction ............ 901 
Prof. P. Lenard on the Electricity of Mercury-falls and on 
Ream TOMS sera. 2, Aeiiel Mawel tie dae. Ad ORO es 903 


Prof. H. A. Wilson on the Statistical Theory of Radiation.. 904 


NUMBER CXX.—DECEM BER. 


Dr. R. D. Kleeman on the Attraction Constant of a Hilal 

of a Substance and its Chemical Properties............ 905 
Mr. D. C. H. Florance on Primary and Secondary y-Rays.. 921 
Dr. J. W. Nicholson on the Approximate Calculation of 


Bessel Functions of Imaginary Argument ............ 938 
_ Prof. J. H. Jeans on Non-Newtonian Mechanical Systems, 
aniualamek s: Wheory of Radiation... ....002.5. <6. 0 ea. 943 
Dr. R. W. Boyle on the Volatilization of Radium Emana- 
sone at OY OE CHA DCPAUUECS a lcinca)t = + oo <ba cdeyese eg eqs Ges 955 


Mr. T. H. Blakesley on a Means of Measuring the Apparent 
Diameter of the Pupil of the Eye, in very feeble Light .. 966 

Mr. Manne Siegbahn on the Study of Variable Currents by 
means of the “ Phaseograph.” (Plate XIX.).......... 969 


Vill CONTENTS OF VOL. XX.—SIXTH SERIES. 


Page 

Prof. O. W. Richardson on the Posit’ve Thermions emitted ‘ 

by the Alkali Sulphates. (Plate XX.)........ 00. name 981 
Prof. O. W. Richardson on the Positive Thermions emitted 

by the Salts of the Alkali Metals ............. one 999 

Lord Rayleigh on the Problem of the Whispering Gallery . 1001 

Prof. J. A. Ewing on’ Magnetic Hysteresis............% 1005 


Notices respecting New Books :— 
Prof. H. C. Jones’s Introduction to Physical Chemistry. 1006 


Annuaire pour lan 1911... Pee 1007 

Mr. G. W. de Tunzelmann’s Treatise on Electrical 
Theory and the Problem of the Universe.......... 1007 
U.S. Coast and Geodetic Survey ..........5 2.0400: 1008 
Proceedings of the Geological Society.............. 1008-1009 
Si re ny SN 1010 


PLATES. 

I. & II. Illustrative of Mr. J. Satterly’s Paper on the Amount of 
Radium Emanation in the Lower Regions of the Atmosphere 
and its Variation with the Weather. 

ILI, Illustrative of Profs. O. W. Richardson and H. L. Couke’s 
Paper on the Heat developed during the Absorption of 
Electrons by Platinum. 

IV. Illustrative of Prof. A. P. Chattock and Mr. A. M, Tyndall's 
Paper on the lonizing Processes at a Point discharging in Air. 

V. Illustrative of Mr. R. T. Beatty’s Paper on the Production 
of Cathode Particles by Homogeneous Réntgen Radiations, 
and their Absorption by Hydrogen and Air. 

VI. Illustrative of Prof. Silvanus P. Thompson’s Paper on Hys- 
teresis Loops and Lissajous’ Iigures. 

VIL-IX. Illustrative of Prof. E. H. Barton and Mr. T. F. Ebblewhite’s 
Paper on Vibration Curves of Violin Bridge and Strings. 

X. Illustrative of Dr. W. H. Eccles’s Paper on the Energy Rela- 
tions of Uertain Detectors used in Wireless Telegraphy. 

XI. Illustrative of Mr. A. E. Garrett's Paper on Positive Electri- 
fication due to Heating Aluminium Phosphate. 

XII. Illustrative of Mr. G. H. Berry’s Paper on the Pianoforte 

-  Sounding-Board. 
XIII. Illustrative of Prof. Taylor Jones and Mr, Roberts’s Paper 
on Musical Are Oscillations in Coupled Circuits. 
XIV. Illustrative of Prof. R. W. Wood’s Paper on a New Radiant 
Emission from the Spark. 

XV. Illustrative of Dr. G. W. C. Kaye’s Paper on a Method of 
Counting the Rulings of a Diffraction Grating. 

XVI. Illustrative of Prof. A. Trowbridge’s Paper on a Vacuum 
Spectrometer. 

XVII. Lllustrative of Prof. R. W. Wood’s Paper on the Echelette 
Grating for the Infra-Red. 

XVIII. Illustrative of Dr. W. Makower and Mr. E. J. Evans’s Paper 
on the Deflexion by a Magnetic Field of Radium B on recoil 
from Radium A. 

XIX. Illustrative of Mr. Manne Siegbahn’s Paper on the Study of 
Variable Currents by means of the ‘‘ Phaseograph.” 

XX. Illustrative of Prof. O. W. Richardson’s Paper on the Positive 

Thermions emitted by the Alkali Sulphates. 


———_ ee a ae lle 
« 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES. 


jif 


JULY 1910) » 


I. Onthe Amount of Radium Emanation inthe Lower Regions 
of the Atmosphere and its Variation with the Weather. «B. 
JouN SatreRty, A.R.C.Sc., B.Sc., B.A., St. John’s Colleg 


Cambridge *. 
[Plates I. & If.] 


‘he the Philosophical Magazine of October 1908 L gave an 

account of the methods I had employed for measuring 
the amount of radium emanation in the air. Two methods 
were used: (1) absorption of the emanation by coconut 
charcoal, and (2) condensation of the emanation by liquid air. 
The first method is the more easily performed and gives the 
more accurate results. In the same paper I described my 
attempts at finding how the amount of radium emanation in 
the air was affected by weather conditions. The experiments, 
however, were not sufficiently sensitive to give any definite 
results, and the purpose of the following paper is to describe 
some more accurate experiments made in 1909. 

Full experimental details of the charcoal method are given 
in my former paper, but it may be worth while to recapitulate 
them here as briefly as possible. 

To get the emanation out of the air, the air was drawn by 
a water-pump through coarsely-powdered coconut charcoal 
packed in a porcelain or silica tube. In all the experiments 
the air-stream was kept at a constant value (half a litre 
per minute), measured by means of a gauge, and regulated 


* Communicated by Sir J. J. Thomson. 


Phil. Mag. S. 6. Vol. 20. No. 115. July 1910. B 


2 Mr. J. Satterly on the Amount of Radium 


by screw-clips on the indiarubber connecting tube. The 
air-stream was continued for a certain time, and from a 
knowledge of the time of exposure and the strength of the 
air-stream the volume of air drawn through could be 
calculated. At the end of the exposure the tube was taken, 
placed on a gas-furnace, and connected up to an aspirator 
formed of two bottles connected by rubber tubing, one of 
which was filled with water. The tube was heated to redness 
and the aspirator then started. The emanation absorbed by 
the charcoal during the “exposure ” to the air-stream was 
driven off by the heat of the furnace, and drawn by the 
current of air into the aspirator. Two aspirators were used 
to make sure that all the emanation was withdrawn from the 
charcoal. In practice, it was found that the air in the first 
aspirator so collected contained nearly all the emanation 
absorbed ; the air in the second one usually contained just 
a little or none at all. 

The amount of emanation in the air in the aspirators was 
measured by the increased electrical conductivity it imparted 
to the air. To do this it was necessary to pass the air into 
a testing vessel. 

The testing vessel consisted of a brass cylinder 40 ems. 
long and 10 ems. wide. Down through the centre of the 
vessel there passed an insulated brass wire whose upper 
extremity was connected to the insulated quadrants of a 
Dolezalek electrometer. The needle of the electrometer was 
kept at +80 volts, and the sensitiveness was such that 
1 volt of the insulated quadrants gave a deflexion of about 
950 millims. on a scale 2 metres distant. The testing vessel 
was kept at —-320 volts. This ensured saturation for all the 
leaks I had occasion to measure. The normal air-leak was 
from ‘9 to 1°2 cms. per minute, it being practically constant 
on any one day but varying from day to day. There were 
two openings into the testing vessel: the upper one led to a 
mercury manometer, and the lower one served as a means for 
exhausting the vessel and refilling it with the air whose con- 
ductivity was to be measured. 

The normal air-leak having been taken, the vessel was 
exhausted, and the air collected in the second aspirator was 
passed into the vessel*. The leak was again taken. The 
difference between this leak and the normal air-leak is a 
measure of the emanation in the air collected in the second 
aspirator. The testing vessel was again exhausted and the 

* The volume of air collected in the aspirator was just a little less 


than the volume of the testing vessel. Thus all the air in the aspirator 
could be passed into the testing vessel, 


Emanation in the Lower Regions of the Atmosphere. 3 


emanation-charged air contained in the first aspirator was 
passed into the vessel. The leak was again taken. 

Now when radium emanation is passed into a testing vessel 
the leak in the vessel does not remain constant. The active 
deposits Radium A, B, C, &., are deposited on the walls of 
the vessel, and the total activity of the contents of the vessel 
increases for about three hours, after which it gradually 
decreases, very slowly at first, and then at the rate of the 
decay of the emanation (half value in 3°86 days). Fig. 1 
shows the curve of activity obtained in one particular case 
(Noy. 6, 1908) from the emanation driven off from the 


Fig. 1. ; 
aS Soe 
ae 


2-30 Ey 5-30 


30 / 4°30 Pag 

TIME 
Curve showing how the leak in the testing vessel after it has been 
filled with air containing radium emanation varies with the time. 


a 


ES SSS ASASSASAASSSS 


Activity (leak) in ems. per min. 
NLEAK & 

Sie liters 

SAAS 


~ 


SSSSS9 


6 SSA 


charcoal after an exposure to the air. It takes about 
3 minutes to pass a bottle full of gas into the vessel. This 
is indicated by the shaded area. Some observers make a 
point of taking the leak at the time of maximum activity 
which occurs after the emanation has been in the vessel 
3 hours. There is, however, as Soddy and Mackenzie* first 
peinted out, a temporary maximum which occurs after the 
emanation has been in the vessel about 10 minutes. The 
leak is practically steady for about 15 minutes after this, and 
much time is saved if the reading of the leak is taken during 
this interval. This is especially true if it is required io 

_  * Phil. Mag. Aug. 1907, 

B 2. 


ee ee 


4 Mr. J. Satterly on the Amount of Radium 


measure another batch of emanation quickly afterwards, 
owing to thesmaller quantity of excited activity deposited on 
the interior of the can, and consequently lesser time taken by 
this to decay into insignificance. 

In all cases, therefore, my readings of the leak were made at 
this first maximum. The leak due to the air from the second 
aspirator added to the leak due to the air from the first 
aspirator full, gives the total leak due to the emanation 
collected from the charcoal. This is not, however, all col- 
lected from the air, as the charcoal itself contains radium, 
and if left to itself gradually accumulates radium-einanation. 
From the total leak mentioned above must be subtracted the 
leak due to the emanation produced from the charcoal itself, 
and the remainder is a measure of the emanation absorbed 
from the air by the charcoal during the exposure. 

The experiments in 1908, the results of which are given 
in my former paper, were carried out with coconut-charcoal 
contained in porcelain tubes. These tubes were 2 feet long, 
and the central foot of the tube was filled with coarsely 
powdered charcoal kept in place by asbestos wads. The ercss 
section of the tubes was 1°8 sq. cm., and a foot-length of 
tube contained 39 germs. of charcoal. The amounts of ema- 
nation absorbed were small, and to increase the accuracy of 
the work it was decided to work with tubes of larger bore. 
In the 1909 experiments opaque silica tubes were used of 
cross section 8°0 sq. em. ; a foot length of these tubes con- 
tained 155 grms. of charcoal. The charcoal was kept in 
place by spirals of wire gauze. The air-stream being kept 
the same in the two series of experiments, a greater proportion 
of the emanation in the air would be absorbed with the silica 
tubes than with the porcelain tubes. 

In the 1908 experiments drying agents (calcium chloride 
and strong sulphuric acid) were used to absorb the water 
vapour contained in the atmospheric air drawn through the 
tubes. The absorptive power of charcoal decreases as the 
charcoal gets wet. Another reason for drying the air was 
that if the air was not dried the charcoal absorbed the water 
from the air ; and when the porcelain tube was heated the 
water was given off and condensed on the cold portions of 
the tube beyond the furnace, often causing the tube to crack. 
There is, however, an objection to drying-agents in that they 
theinselves may give off or absorb radium emanation. In the 
present series of experiments no drying-agents were used. 
Silica tubes will not crack under the conditions mentioned 
above. Also, with such a large quantity of charcoal the water 
caught will be lodged in the first portions of the charcoal 
column which the air meets, and there will be plenty of dry 


Emanation in the Lower Regions of the Atmosphere. 


charcoal behind to absorb the emanation. Hxperiments, 
however, were made to test whether the amount of emanation 
absorbed depended upon the dryness or wetness of the air, 
and it was found to be practically independent of the amount 
of moisture present. 

In the present series of experiments to measure the amount of 
emanation in the air, two silica tubes each containing the same 
amount of charcoal were used, and they were laid side by 
side upon the bench. The left-hand end of each tube was 
Joined to the side-arms of a T-piece (see fig. 2), the leg of 

Fig. 2. 


the T communicating by a long glass tube to the outer air, 
its extremity being about 6 feet above the ground. The 
right-hand ends of the tubes were joined to gauges which 
measured the air-stream, and these communicated with a large 
bottle from which a pipe led to the water-pump. During the 
experiments, air-streams of the same magnitude (half a litre 
per minute) were drawn through the two tubes; so that if 
the charcoal in the tubes is in the same condition as regards 
size of particles and packing, the tubes ought to absorb and 
consequently to yield when heated the same amount of 
emanation. Usually, as will be seen from the tables 
(pp. 10-17), the amounts were very nearly equal; but in a 
few cases there were unaccountable discrepancies between 
them. At any rate one tube served as a check on the other. 

The tubes having been first cleared of emanation by heating 
to a red heat and drawing air through them between 2 and 
4 P.M. were connected up as described above, and the air- 
stream started at about 5 p.m. After an hour’s reading of 
gauges and regulating of clips the air-streams were obtained 
steady at half a litre per minute, and the tubes were left for 
the night. At about 8 4.M. next morning the gauges were 
read ; and if there had been any wandering away from the 
mark the air-streams were again adjusted to half a litre per 
minute. Justabout 2 p.m. the gauges were again read and the 
air-stream stopped. The exposure had thus lasted for about 21 
hours, and about 630 litres of air had passed through each tube. 

The tubes were then heated in turn, and the emanation was 
tested as described above. The heating and testing took 
from 2-6 P.M., but the tubes were cold by 5 P.m., and they 
were then reconnected up and the air-stream set running for 
the next day’s reading. 


6 Mr. J. Satterly on the Amount of Radium 


The Correction for the Growth of Radium Emanation 
in the Charcoal. 


As explained above, deductions must be made from the 
observed leaks given by the gas passed into the testing vessel 
for the leak due to the emanation generated by the charcoal 
itself since the last heating. 

If at a heating the charcoal is completely deprived of its 
emanation and then the tube is closed up and left to itself, 
the emanation will gradually accumulate, the equation of the 
production being | 


iF — I,(1 ae oo); 


where I,= the amount in existence at time f, 

I,=the amount in existence after an infinite time, 

e =the base of the natural logarithms=2°71828...., 
and )=the radioactive constant of radium emanation. 

Taking the time for radium emanation to decay to half 
value as 3°858 days*, we have, using the nomenclature of 
Rutherford, 
T=3°858 x 86400 secs. ; 

and since AT=log, 2=°693, 


= 2°083 x 10-* sec.~, 
Therefore the equation for the production of emanation in 


the charcoal is 
T,=1,(1 —e7 2083x107 xt), 
where ¢ is in seconds or 
I,=1,(1—e—"), 
where ¢ is in days. 
From this equation, and taking the value of I, as 100, the 
following table has been calculated fT. 


TABLE I. 


¢(days).| 1. jo. Be Bt fe) TON 252 |) Qian 


The reason why these particular values of ¢ were chosen 
will be seen later (Table II., p. 8). 


* Kolowrat, Le Radium, July 1909, and Curie, ib¢d. Feb. 1910. ) 
+ Tables suitable for these calculations will be found in Kolowrat’s 
paper mentioned above. 


Pe ee 


Fimanation in the Lower Regions of the Atmosphere. 7 


From this table we see that, if after a period of rest of 
six days the amount of emanation accumulated is M, the 
amount that would be accumulated in one day is 


16°5 


66°0 


Throughout the experiments it was the accumulation 
per day that was required. This amount, however, would 
be hard to measure accurately, and to get the daily accumu- 
lation the tubes were allowed to rest for longer periods. At 
the end of such a period of rest, the emanation accumulated 
was driven off by heating and the amount tested in the usual 
way. The usual period of rest was from Friday afternoon 
to Monday afternoon—3 days,--so that Monday’s heating 
expelled a three days’ accumulation. Table II. (p.8) gives the 
results obtained for this accumulation. The fifth column 
was obtained from the third and fourth columns by using the 
figures in Table I. as described above. 

The figures obtained for the daily accumulation of A or B 
are not quite constant. This is inevitable from the smallness 
of the quantity under measurement ; the agreement, however, 
indicates that it is radium emanation that is being generated. 
The average value of the leak due to the daily accumulation 
works out at -46 for tube A and °48 for tube B, or approxi- 
mately °5 for each tube. 

It was just possible that some of this accumulation might 
be driven out by the air-stream during an “ exposure,” so 
that from the reading of the total emanation collected from 
the heated tubes after a day’s run less than *5 should be 
deducted. It is rather difficult to test this point accurately; 


| but experiments made by giving the tubes just an hour’s 


“ exposure ” showed that none of the accumulation had been 
driven out in that time, so that perhaps all the accumulation 
remains proof against a twenty-one hour air-stream. 


As a sample reading I will quote that of Thursday—Friday, 
Aug. 10th—11th. 

Air-stream started through tubes at 5.14 P.M., gauges 
checked at 8.4 A.M. next morning, again read at 2, 14 P.M., 
and current then stopped. The stream had remained practi- 
cally at °5 litre per minute all the time, A had gone up 
slightly, B down. ‘The time of exposure was 1260 minutes, 
and the gauge readings showed that 635 litres of air had 
passed through A and 625 litres GE BL 


8 Mr. J. Satterly on the Amount of Radium 
TaBLeE II. 

Amount accu- | Amount accu- 

Date of Charcoal | Period of | mulated in this} mulated per 

Testing. Tube. Rest in |period expressed| day expressed 

days. as cms. per as cms. per 

1909. min. leak. min. leak. 

Mon. Feb. 1 A 3 8 "32 
ee: Veneee A 6 23 ‘oT 
Wed. Feb. 10 ... B 6 2°9 af 
Mon. Feb. 15 B 3 m1 *44 
" . A 6 Li "28 
Tues. Feb. 23 A + 1:3 "42 
> .» B z 8 *26 
Mon, Mar. 1 B 3 1°4 56 
dy - 2s A 3 1:2 "48 
Mon. Mar. 15 .. A 5 1°4 39 
" ‘5 - B 5 2°1 "59 
Mon. Mar. 22 .. B 3 1:3 "52 
Mon. Mar. 29 .. B 3 11 “44 
> ” “ A 3 1-2 "48 
Mon. Apr. 26 ... B 25 13 “30 
i - A 25 17 ‘28 
Mon. May 3 B 3 10 40 
ye A 3 11 ‘44 
Mon. May 17 B 3 31 1:2* 
A : un A 3 53 2 iF 
Mon. May 24 .. B 3 1:4 "56 
" ‘5 A 3 1-4 *56 
Mon. May 31 B 3 1-2 "48 
* 4 A 3 1:3 52 
Mon. June 7 B 3 I "44 
¥, n i A 3 1:3 02 
Mon. June 21 .. B 3 10 “40 
y } A 3 15 ‘60 
Mon. July 19 B 27 2°6 43 
Mon, Aug. 2 B 10 19 “37 
a is se A 10 2°4 "47 
Tues. Aug. 10 .. B 4 15 "48 
“ i ss A 4 19 “61 
Mon. Aug. 23 .. A 3 18 vg) 
= ‘ : B 3 12 °48 
Thur. Oct. 14 ... A 48 2°8 “47 
B 48 2°7 *45 


” %” 


* Very high. Rejected when firding the mean value. The tubes had been 
opened up and the supply of charcoal replenished on May 15. 


Procedure of Testing. 


(1) Exhausted the testing vessel with an oil-pump, refilled 
with dry air, took the air-leak : leak=1:0 cm. per min. 
(2) Meanwhile heated tube A on the furnace, and collected 
in succession two aspirators full; call them aandd. ~ 


Emanation in the Lower Regions of the Atmosphere. 9 


(3) Exhausted the testing vessel, passed in the contents 
of 6, took the leak: leak=1°1 cm. per min. 

(4) Exhausted the testing vessel, passed in the contents 
of a, took the leak : leak=4°8 cm. per min. 

Hence we get for Tube A: 


“a 120-1 | 
Ag Do | 3-9 
6 4:8—10=3'8 

The tube had been heated the day before, so taking away 
‘5 for the day’s accumulation we get left 3-4 due to the 
emanation absorbed by A from 635 litres of atmospheric 
air. 

(5) Exhausted testing vessel twice in succession, refilling in 
with dry air each time. This was to clear out all the ih 
emanation due to A(a). The leak was now high due to the | \\ ) 
active deposit from the emanation of A(a). The activity 
of this deposit gradually decayed, and after about an 
an hour the leak was fairly steady at 1-1 cm. per min. 

(6) Meanwhile heated tube B on the furnace, and collected 
(a) and (0) as before. 

(7) Hxhausted the testing vessel, passed in 6 : leak =1°3 cm. 

er min. 

(8) Exhausted the testing vessel, passed in a: leaak=5°3 cm. 

per min. 


Hence we get for Tube B: 
ry ee oe 
B ake 4-1 
G “ ig ay 

Take away ‘5 for the accumulation since yesterday and we 
E. get left 3°6 due to the emanation absorbed by B from 

625 litres of atmospheric air. 
(9) Exhausted testing vessel twice and refilled with dry 

air each time to clear out all the emanation due to B (0). 


(10) Meanwhile set A and B running for the next day’s 
results *. 


Results. 


The following tables give my results. Owing to the 
closing of the laboratory at the week-end it was not possible 

_ to take more than four daily readings in succession. 
Careful notes were made of the weather phenomena, such 
as the height of the barometer, the direction and force of the 
wind, &c., while the experiments were in progress. Most of 


* As this particular set of readings was taken on a Friday, A and B 
were not set running but were left to stand till the following Monday. 


Date 
1909. 


ae 


Mar. 


Mar. 


Mar. 


Mar. 


Apl. 


Apl. 


{Eup 


fet bed feet bed 
WwrOOM-] 
noes 


10 


Mr. J. Satterly on the Amount of Radium 


TaBLeE III. 
Wind 
: Thermo- | Humidity. | Direction and 
oe meter. i Force. a |S 4 Weather Remarks. 
9 a.m.|9 p.m.|Max,|Min.| 9 a.m.) 9p.m.] 9 a.m. | 9 p.m.] oS ae 
mm.) mam, | OR, | OR, in. 
758 | ‘757 | 86 | 29 | 83] 100 | SE1 |NE2-3) ... | 5:5] Sunny 9a-3p. 
756 | 752 | 85 | 80 | 100} 87 | NE2/ N 1 |-02 | 0:3! Snow. Sleet from 5p. 
747 | 743 | 88 | 21 | 100| 84}]SW1 | SE2 | -29 | 0:2] Fog 9a. Overcast. 
740 | 742 | 34 | 28 | 100} 83 |SE2-3] E2 |-02 | ... | Snow? night. Snow showers 
745 | 749 | 85 | 18 | 97 | 100 |NW2/|Calm0O| ... | 5-5} HazeQ9a. Fine and sunny. 
foo). 700 | 40) 12) 100 | S8 SWI), Ooi) 6-4 09 0 ” 
745 | 737 | 37 | 25 | 100 | 97 | SE 4 |SSE5-6/-62 | ... | Snow? 7a-3p. 
[sunny. 
737 | 745 | 44 | 33 | 100 | 100 |SW2 |SSE2}-10 | 3:0| Snow 6a-9a. Fine and 
751 | 754 | 48 | 31 | 92] 96 | SE2|SE2| ... | 9-6| Fine and sunny. 
753 | 753 | 28 | 33 | 95) 97 INW1 |Calm0/:15 | ... | Snow. Rain 8a-3p. 
754 | 754 | 42 | 33 | 94] 92 |NE1 | E5-6| ... | ... | Overcast, with haze. 
755 | 758 | 86 | 34) 92| 93 |NE4| N 41-01 | ... | Overcast, with rain. 
760 | 759 | 40 | 34| 90) 91 INNW4;NW2| ... | 1:3} Overcast till 3p. 
757 | 752 | 40 | 32 2 | 100 |NW8-4/Calm 0} -17 | 3:0} Sunnya. Snow? 3 p. 
744 | 742 | 88 | 31 | 80] 98 |W+45|NW 5}-10 | 0:1} Overcast, with snow. Cold. 
741 | 745 | 36 | 28 | 89| 100 |NW2 |NW2|-08 | ... ; ; d 
748 | '749 | 41 | 25| 95] 95 | W2 |\Calm0O) ... | 6-3| Fine and sunny. Snow °. 
750 | 750 | 45 | 22] 91] 90|SW1|SH2| ... | 7-1] Fine and sunny. 
746 | 742 | 47 | 33 | 92] 94] SE3| SE2 |-02 | 0-4] Overcast, wet. 
741 | 744 | 56 | 42] 97) 96 | SE2/| SE2 |-03 | 3:3] Sunny intervals. 
747 | 743 | 56 | 42) 95| 99} S2] S1 |-06 | 1:5] Sunny intervals till 2 p. 
Rain 4 p. 
750 | 749 | 53 | 41 | 81 | 100 |SW3-4| SE1 |-06 | 62| Rimem. Fine and sunny. 
752 | 751 | 49 | 42 | 98] 98 | SE1 | SE2|... |}... | Fog? from6a. Overcast. 
754 | 758 | 48 | 42 | 99] 100 | NE2 |Calm 0) ... | ... | Fog? from 6a. Overcast. 
755 | 743 | 51 | 88 | 94} 94 |S82-3 |[SW4-5/-39 | ... | Overcast. Rain? 2 p-11 p. 
739 | 741 | 54 | 42 | 80| 82|SW6 |NW6/-03 | 2:4| Rain? showers. 
747 | 753 | 48 | 87 | 78| 88 |INW6 |Calm 0! ... | 4°9| Sunny intervals. 
756 | 755 | 52 | 31 | 79) 81 | W2 |Calm0}-01 | 5:5} Fine and sunny. 
749 | 744 | 52 | 37 | 92| 92] SE3| S34 |[-16| ... | Rain most of day. 
742 | 740 | 60 | 47 | 79| 98| S38 | S2 |-02 | 4:1| Fine. 
740 | 746 | 52 | 46] 92] 87| S2|SW2|-44| ... | Raina & Rain? 1 p-5p. 
750 | 747 | 55 | 45 | 69| 94 |SW2|SW4/-02 | 4°8| Fine and sunny a, 
758 | 769 | 50 | 389 | 84] 72) N3| N1 |... | 5:2| Sunny intervals. 
773 | 774 | 47 | 29| 66] 86] N2 \Calm 0 3°3| Cloudy. Cold wind. 
774 | 773 | 48 | 28 | 68| 76 | SE3 | SE3 9-4| Cold wind. 
773 | 772 |, 51| 32 | 66| 77 |SE2 | SE1 1... J12-0] Fine and sunny. 
757 | 753 t Lovely weather. No rain, and average'11 hrs. sunshine each day. 
&,M, Tv, Fine. W, windy. Tn, NW, fine. F & 8, SW, showery. 
757 


797 | Fine and sunny. Rest of week-the same with a little rain. 


Emanation in the Lower Regions of the Atmosphere. Lal 


TasLe ILI. 


sane ’ 
een es Weekly Weather Notes (abbreviated from the Weekly Weather Report 
a of the Meteorological Office). 
-|Tube A|/TubeB 


a | - 


= Weather, wintry. Barometer and Wind: The general distribution of pressure 
vi favoured winds from some point between N & E. In addition to a large cyclone 
Dy of irregular form that extended over this country early in the week a well defined 


L2-8 26 system moved southward over Britain and the North Sea during Tu and W, and 

finally disappeared in the Netherlands. On Friday a band of relatively low 
Be f2:8 2°6 pressure extended from the Atlantic over the N part of these islands and winds 
.6-7* 4-4 became W. But on Saturday the advance of a deep cyclone to our S.W. coasts 
drew wind into NE & SH. 


D Weather, unsettled. Barometer and Wind: At the beginning of the week a 
i depression had arrived and lay over England causing complete cyclone circu- 
WY lation. This system moved SE across the Channel. The barometer meanwhile 
Vv stood highest in Iceland and N. Hurope, but later in the week the pressure gave 
a vee ue cael and became high out West. Wind changed from NE to 
: and increased in force. 
) 
a Weather, unsettied. Temperature, low at first but rose later. Barometer §& Wind: 
| t co. | 16 At the commencement of the week a depression covered the North Sea and the 
Tu general winds were N. ‘This continued till Tuesday when a shallow low- 
W bar] 4:0 | pressure system appeared between Scotland and Ireland, and wind backed to W 
lo9 | 20 & SW. This was followed by another cyclone irom the Atlantic, deep at first 
et |). but which got shallow as it travelled over U.K. The wind after blowing 
? b2-4 20 strongly SH on our S.W. coasts veered to S and became light. 
q 
a Weather, unsettled. Barometer and Wind: During nearly the whole of the week 
} 19-0 2:9 depressions either extended slowly or passed directly over U. K. while the 
vu barometer was high in Iceland and Spain. The centre of the chief system passed 
WW bo-4 22 over Yorkshire Wednesday night. As it increased considerably in depth 
Bio. | 1-2 during its passage the wind blew a strong breeze or high wind between SW 
[TH |) & S on the south coast and E & NE in the north, while in the rear on Friday 
7 19-0 16 a NW breeze was general. On Saturday another cyclone was approaching. 


> | Weather, unsettled, finer after Wednesday. Barometer and Wind: During the 
it 128 2-9 greater part of the week depressions passed across U. K. while the barometer 
uy was high in Iceland and Spain. After a rapid passage of one on Wednesday 
W 22 18 an anticyclone set in from the north, and the barometer rose rapidly. This 


16 16 increased in size and intensity as it passed across the North Sea on Saturday. 
a With the cyclonic distribution the wind was W & SW in the south and 
: 3 2°5 strongly E in the north. 


{" 


fan 27 23 


* Decay curve taken and its identity with that of radium emanation verified. _ 


Apl. 25 
26 M 
27 Tu 
28 W 
29 Th 
30 F 


May 18 


May 


= Up 


eae 


DOIG OUP OO 
mie 


May 95 


10 M 
LEsio 
12 W 
13 Ta 
14 F 
15 8 


May 16 } 
17 M 
18 Tu 
19 W 
20 Tu 
21 F 
22°85 


Barometer. 


9 a.m.i9 P.M! 


~J J I 1-1 
ROA 
NSS To ES ome) 


fo) 


Mr. J. Satterly on the Amount of Radium 


Beg ae 


60 


66 
59 
60 
57 


3 ye 


51 | 


Thermo- 
meter. 


on 
43 
44 
46 
38 
39 
38 
34 


| 


TABLE IV a. 


Humidity. 


-—- 


78 | SW 6 


69 
82 
86 
97 
88 


93 


Wind 
Direction and 
Force. 


Max. Min.|9 ena P.M.| 9 a.M. | 9 Pim. 


S 2 


SW 2] SwWl 
SW 3-4] W 2 
WSW 4) SW 5 


W 4 |Calm 0}: 


NW 4] w5 
NW 4| SE2 


NW 4] SE 2 


N 5 N 4-5 
N 2-35 | Calm 0 
SW 2-3) SW 2 
W 2 N4 
N 4 N 2 
Calm 0| Calm 0 
N 2-3 1 | 


SE4 |N4-5 


W 3-4!Calm 0]: 


SW 5-6 W 2 
SW 1 |Calm 0 
S 2-3 | SH2 
ESE 2} Calm 0 
W 1 |Calm 0 


S 
a 


| Hours of 
| Sunshine 


Weather Remarks. 


—~ - 


Sunny. Thunderstorm at 
[noon. 

Continuous sunshine. 
Fine. Rain and hail at 7 p. 

Sunny till 33 p. 

Sunny5a-2p. Rain? after 
[4 p. 
Sunny 
[day. 
Fine and sunny. Snow,® 
[rain® 6 p. 


Snow® and rain® a. 


Haze a. Sunny. Sleet at} 
[noon.| | 
Fine and sunny. ) 
Continuous sunshine. 


Haze a. Sunny 5-9, then 
[cloudy. 
Overcast, brighter evening. 
Fine and sunny. 
Overcast from1p. Rain6p.| 
Sunny intervals, damp. | 
» chilly, damp.|, 
Sleet 10a, damp. 


be) 
Rain®. 


Fine and sunny, chilly. 
Overcast. Rain® 3 p-5p. 
Sunny intervals. 

Fine and sunny, hot. 


Emanation in the Lower Regions of the Atmosphere. 13 


TasBe IV a. 
Emanation 
oo, a fee Jitres Weekly Weather Notes (abbreviated from the Weekly Weather Report 
i of the Meteorological Office), 

Tube A Tube B 
2 Weather, unsettled. Barometer and Wind: Throughout the week the 
M barometer was relatively high in Icelind and Spain while cyclonic 
Ty disturbances either extended or travelled across U. K. ‘These systems 

bog 1:9 were not asa rule deep, and the wind which was mainly between S and W 
WwW bo. ; had little force. In the rear of a moderate depression which passed 
TH 20 | 19 across the North of England on Thursday, the barometer rose briskly 
F b1-9 14 and the wind veered to NW or N over the whole of U. K. and increased in 

force. 

8 
2 Weather, dry, cloudless. Barometer and Wind: Distribution mainly 

16 1°4 anticyclonic. At first the pressure maximum was over France, and it 

U travelled from there to Germany, and to Scandinavia. uring the latter 

Sy - lled f h G y, and to Scandinavia. During tbe | 
Ww {25 18 part of the week a large cyclonic system appeared over the Bay of 

118 1:8 Biscay. This was accompanied by an E wind over England and a SE 
Tu wind over Scotland At the end of the week a well-marked but not 
EF b 1-5 16 deep depression travelled southward over Scandinavia and highest 
z pressure existed in Iceland. 
be) 
2 Weather, fine. Barometer and Wind: At the beginning of the week the 
(66 58 barometer was highest in Iceland and lowest to 8.W. of U.K., and in 
Tu ay 3-8 Germany. A moderate NE breeze was blowing. On Monday the anti- 
Ww ; eyclone extended over these islands. Later it moved NW to the 

t17 16 | Atlantic and a depression born in the Shetlands travelled to Finland by 
Tu 2] 24 Thursday. After this small disturbaices travelled southward over U. k. 
F t and the North Sea. The general direction of the wind was between 
‘ _ N and NW sometimes W and sometimes E. 
s 
= noe fine and dry. Barometer and Wind: The distribution of pressure 


3 6 | 36 underwent several changes though the barometrical movements were not 

brisk. Pressure which was highest over Iceland gradually gave way 

{84 | 78 | while it was sometimes highest to the S., E., and N.E. of U. K. No 

13-0 36 important depressions passed over U. K., but towards the end of the week 

: a large system was spreading in slowly from the Atlantic. Wind, light 
{ 2-4 26 | and variable except in latter half of week when it was S & SW. 


Mean ot 29 


4 


May 23 5 
24 M 
25 Tu 
26 W 


31 To 


WwW 


14 


Barometer. 


9 a.m.|9 P.M. 


mm. 
764 
770 
754 
749 


751 
758 
799 


766 
763 
757 
760 
764 
756 
799 


758 
762 
762 
763 
762 
759 


760 


765 
767 
768 
766 
768 
770 
167 


761 
754 
744 
748 


mm. 


Mr. J. Satterly on the Amount of Radium 


Thermo- 
meter, 


Max. 


Min. 


Or 
50 
42 
dl 
45 


49 
47 
50 


46 
53 
51 
49 
46 
46 
45 


TABLE IV B. 


Wind 
Humidity. | Direction and Se 
Korce. a |S a 
See 
- 128 
9 a.ul9 pw} 9am. | Opa. | i A 
in, 
59 76| W1 |NNW3-4 ... |10°8 
55 62 N2 SE 4 | °34/13:0 
98 | 99 |SE 2-3 Ss a ory 
69 | 85 |SE5-6|SW 6 | 45] 4:7 
74 | 98 |SW5 |Calm 0| :05| 3:8 
65 74 |WSW 2} SW 4] ‘01)] 9:5 
65 Sl | W 4-5 | Calm 0} <.. 111°7 
71 91 |ISW 2-3) Calm 0! ... {103 
78 86 S2 |Calm 0| ... | 9°3 
81 | 100 INNW 4; N4 | ‘69! 0-2 
93 85 N 4 INA OT oe 
71 79|NE5 |NNE 5| :07| 4:3 
92; 93 | NE 4 Noo! Roe eae 
94; 83 |NW2 Ne 
78 97 | NE 1 |Calm0| ‘04| ... 
AO ae Ni?) NR ea: 
72 bow 2 Wl ae 
79 GMa IN 4 
80| 85!|NE4 |NW1 | :02| 1:7 
65 77|N45 Wid 222) 162 
97 80 | NW 4 INW 5-6] ‘01} 1:0 
74| 78|NW4 N 4 13 
70; 80| NW1 | NEI 10°5 
72| 94 W2/|NE2 32 
89 | 86| NW3 N 4 2-9 
75 | 84 IN SE) | NS 6°5 
87 88 | NW3 | Calin 0 Dt) 
78 86 | SW11SW 2 03 
79| 82| SW5|SW4|... | 4 
77 86 |SSE 4-5| SSE 1 | 02] 3:5 
84 | 74 85 $4 VP Coo 
64 | 80] SW 6 | SSE 2 | :22| 9:1 


Weather Remarks. 


Sunny intervals. 

Fine and sunny. 

Rain? 7a—noon. Overcust. 

Rain? 11 a-1l p. Thunder- 
[storm 2p -3p. 

Showers. Sunny intervals. 

Sunny intervals, 

Fine and sunny. 


Fine and sunny. 
Sunny intervals. 
Overcast. Rain?. 
Rain till noon. Overcast. 
Sunny a., then overcast. Cold. 


Glorious. 


Rain all day. Cold. 
Foga. Overcast. Cold. 
Wet. Overcast. Cold. 
Haze. Fine and sunny. 
” ” ” | prove. 
Overcast till 4p. Then im- 
Z » Op Cold 
Sunny till noon. Rain. 
[Sleet 14 p. 
Rain? 7a-8a, Cloudy. 


Fine and sunny. 


Overcast till 34 p. 
Cloudy. Cold. Dull. 
Haze. Sunny 9a-3 p. 
Sunny 10a-4p. 
Overcast till 3p. 


Sunny a. Thunder 5p. 
Cloudy. Dull. Rain 2p. 
Rain 10a-noon. Sunny 2p-7p 
Sunny till 1.30 p. : 


Emanation in the Lower Reyions of the Atmosphere. 15 


Taste IV ps. 
_Emanation | 
ee) Stes Weekly Weather Notes (abbreviated from the Weekly Weather Report 
ay. of air. 
of the Meteorological Office). 
Tube A/Tube B 
> Weather, much less bright. Barometer and Wind: Depressions from the 


3 19 Atlantic of variable size and no great depth, travelled over these islands 
and the pressure changes were consequently frequent and rather consider- 
3 


16 able. The wind was generally light and moderate between SW and W 


Ww 9 9 except in Scotland where it was SSE and E. A strong breeze or high 
wind prevailed for a time on our southern coast on Wednesday. 


Weather, fine and bright, then extremely unsettled. Barometer and Wind : 


(9.8 4-0 In the early part of the week a large V-shaped depression travelled 
Tv |§ eastwards across U. K., and light to moderate breezes were experienced 
Ww 8 8 from S& SW toNW&N. After Wednesday a well defined cyclone 

{ 8 10 spread northwards from Spain and NE breezes set in over all U. K. 
Tn an increase in the strength of the wind occurring on ‘Lhursday. Towards 
R t “9 6 the re J te we oe eee ae cae gradually dispersed and the 
5 wind pac to N and became light. 
= Weather, cloudy, dry. Barometer and Wind: Soon after the week com- 


; 24 ||) O5 menced a large anticylone extended gradually over these islands from the 
Ly. 4 V7 Atlantic, and by Tuesday almost the whole kingdom was under its 
W influence. During the latter half of the week, however, it retreated 

b 1-3 15 very slowly westwards while shallow ill-defined disturbances appeared 


21 1 over theNorth Sea and to the S.E. of England. The general direction of 
B the wind was N and NW but it varied somewhat in places. 
S 
= Weather, dry, cloudy. Barometer and Wind: During the greater part of 
Lo-8 28 the week the barometer was high over the Atlantic and low in Denmark 
Tu : and the general current of air was fromthe North During the end of 
Ww { 19 19 the period the Atlantic system moved to the Bay of Biscay and France, 
‘9 3 and depressions began to affect the north and north-west of U. K., so that 
Tu t 30 | 34 the wind backed to NW and eventually to W and SW. 
F 
Ss 
= Weather, dull. Barometer and Wind: The depression at the N.W. of U. K. 
4 119 19 gradually extended over U. K. The winds were SEH and strong. 
‘U 
Ww 


16 Mr. J. Satterly on the Amount of Radium 


TABLE V. 
_ Wind 
Date Te ae oe Humidity. | ake and Poe 
909 meter. orce. a [og 
1909. | it = P'S Weather Remarks. 
2 lee 
| 9 A.M. | 9 p.m.|Max./Min.|94.m.| 9 p.m.| 9 Am. | 9 P.M. | 93 = a 
Co ae # pena de S peewee J bt ted 
| mime) mms) ° B.)° FB. in. 
|Aug. 1 S| 762 | 759 | 63 | 56 | 74) 99) SE3/ 83 |-59 | 1:0] Rain? noon-4.30p. 
/ 2M! 759 | 765 | 60 | 51 | 94) 88 INW6/NW2}|-03 | 1:0| Rain 8a-noon. 
| 3 Tu| 766 | 766 | 64 | 49 | 63 | 94 |NW 4 [Calm 0} ... | 7-1) Fine. Sunny intervals. 
| 4 W| 766 | 766 | 73 | 48 | 72] 88|SW2 CalmO| ... | 9:3 Fine and sunny, 
! 5 Tu 767 | 768 | 78 | 48} 63) 88} SWI |Calm 0) ... |13:1| Dew. _,, “4 
) wearer) oo | 78 | 51 | 47 | 88 | B21Galm 0 1138) Dew. ae i 
) ein 763.) 79 | 47 |..67.| 95) 281) eed), 124. daw ¥ 
| Aug. 8 &| 764| 763 | 79 | 50| 54 | 72|NE3|NE 2] .., |13-0| Dew. Fine and sunny. 
| 9 M| 762] 763 | 738} 49; 96] 86 | N3 [Calm 0 ... | 8:0] Fog?till lla. Dew. Fine. 
i 10 Tr} 765 | 767 | 74 | 50 | 66); 89 |NEH3 [Calm 0} ... |12°0) Fog? till 7a. Dew. Fine. 
. 1] W| 768 | 766 | 81 | 55 | 94) 92 |SW2 |Calm 0 ... | 9°3] Dew2. Sunny after 10a. 


0 
13 F | 764 | 765 | 75 | 54) 68/ 8L |NW4|NW3| ... | 65| Sunny intervals. Gusty. 
| 14 § | 765 | 765 | 73 | 53 | 73 | 82}SW5|Calm 0) ... | 5-9 


” 99 


. 
12 Tn) 757 | 764 | 85 | 50} 55 | 83 | SW2 |Calm Oj ... |13:0| Fine and sunny. 


Aug.15 | 763 | 760 | 84 | 55 | 74| 88 S 1 |Calm 0} +11 |11°3|} Hot and close. ; 
16 M| 753 | 753 | 74 | 59 | 93 | 82 |SW3/WSW4!/-06 | °3| Thunderstorm at 9a. Rain? 


17 To} 755 | 752 | 70 | 49 | 67} 90} W1| SE4/:14 | 80) Sunny tid 2p. Rain 6-9p. 
. 18 W| 747 | 755 | 69 | 55 | 74!) 85} S83 | SW 4):22 | 60! Thunderstorm 9a~1la. Rain 
| 19 Tn} 759 | 762 | 71 | 55 |} 73 | 76|SW 6) SW 4/-04 |10°0/ Sunny. Rain®, 

20 F | 760 | 755 | 66 | 55 | 87 | 90 S5| SW 5/12] ‘L| Overcast. Rain®. 
| 21S | 753 | 752 | 66 | 49 | 70| 97 | W1 \Calm 0}-14 | 86] Finemorning. Rain?4p-5p. 


jAug.22 &| 752 | 755 | 63 | 45 | 70] 85 | W5 [Calm 0} ‘v1 | 8:8| Fine morning Rain 1} p. 

23 M | 756 | 755 | 67 | 48 | V6| 94 S$4|}SW 4}|:08 | ‘9| Overcast. Rain at intervals. 

24 Tu; 755 | 754 | 67 | 58 | 80| 94 | SW 5|Calm 0) -02 | ... Me Hew showers. 

25 W | 753 | 756 | 68 | 56 | 82) 95 | W4| NW3}|-27 | 6:0} Sunny intervals. Thunderst 
at lla-noon. Rain? 13 p. 

26 Tu} 760 | 762 | 62 | 53 | 89 | 95 |NW4/Calm 0}... | °8| Overcast. 

27 F | 762 | 765 | 67 | 46 | 73 | 84|SW2| W3/... | 46/Sunnya. Overcast p. 


_Limanation in the Lower Regions of the Atmosphere. bf 


TABLE V. 
Emanation 
from 630 litres 
- Of air. Weekly Weather Notes (abbreviated from the Weekly Weather Report 
ca. of the Meteorological Office). 


y. 
Tube A|/Tube B 


S Weather, after first day or two great improvement. Barometer and Wind: 
M _| Soon after the commencement of the week an anticylone of considerable 
t 1-2 1-7 size began to extend over these islands from the Atlantic and it lay over 
Tu 3] 3.7 U.K. and district nearly all the week. On Wednesday a large cyclonic 
Ww system invaded the N.W. ot U. K. but it retreated later. The wind varied 
40 4:3 considerably in direction early in the week: while the highest pressure 
Ta 3-0 3-6 was on the Ocean it was mainly N, and on Wednesday S & SW. On 
& ‘ subsequent days it was W in the north and from the E in the south. 
= | 
M : 
Ty Weather, very fine and bright. Barometer and Wind: During the whole of | 


Ww Lo3 2-3 the period the centre of a big anticyclone lay over or near U.K. Winds 
light and variable. In the extreme north strong westerly breezes occa- 


TH t 55 ; sionally blew, due to an eastward travelling Icelandic depression. 
Ss 


Weather, unsettled, lot of rain. Barometer and Wind: At the commence- 

‘ ment an anticyclone extended over England from the Continent. This | 
To ¢ 21 2-4 soon went E,and Atlantic disturbances travelled eastward over the Icelandic | 
Ww bi-4 15) regions, while their secondaries travelled E-wards & NE-wards directly | 


| 
p [fs | 36 | 
| 


& over U. K. General current of wind was between W & S and fresh at 
Tu b1-4 PD it times, a strong breeze being experienced over our W. & S. coasts. At 
RF b 1-2 1°3 the end of the week a pressure minimum developed over N. Sea, giving 
5 U.K. a NW wind and low temperature. 


16 part of the week the centre of a depression moved eastwards over 
A Scotland, causing E winds in the far north and SW to NW over U. K,. 
3 generally. By Thursday the eastern edge of an anticyclone began to 
13 extend over us from the Atlantic, and this system continued the chief 


| 
Weather, unsettled, frequent rain. Barometer and Wind: During the eariier 
factor over England and S. of Ireland to the end of the week. | 


Phil. Mag. 8. 6. Vol. 20. Now 115. July L910: G 


18 Radium Emanation in Lower Regions of the Atmosphere. 


the weather data in the tables are, however, taken from the 
records given in the Meteorological Office’s publication 
entitled ‘Observations at Stations of the second order and 
at Anemvograph Stations.’ The Cambridge station is at 
the University Botanic Garden, situate about half a mile 
from the laboratory and at the same level. Readings are 
taken there twice a day—9 a.M.and 9 p.m. The maximum 
temperature occurs during the afternoon, so the maximum 
thermometer is read at 9 P.M. and entered to the day. ‘The 
minimum temperature usually occurs in the early morning. 
It is read at 9 a.m. and entered to the day. The humidity is 
obtained from the ordinary wet and dry bulb thermometers. 
The rain-gauge is read at 9 A.M. each day, and since most of 
the 24 hours since the last reading occurs in the previous 
day, it is entered to that day. The sunshine is measured by 
a Campbell-Stokes recorder. The wind force is given on the 
Beaufort scale. It is estimated by the observer using the 
indications given in the ‘ Meteorological Observer’s Handbook.’ 
The aumbers 0, 162, See te 8 on the Beaufort scale cor- 
respond approximately to w rind velocities of 0, 2, 5, 0p 
21, 27, 35, and 42 miles per hour. A number affixed to a 
weather phenomenon under the column Weather Remarks 
indicates the intensity of that phenomenon, thus, Snow°= 
light fall of snow, Rain?=heavy fall of rain. Also a=a.M., 
p=p.mM. The amount of emanation is given as the leak pro- 
duced in my testing vessel measured in cms. per minute on 
my electrometer-scale. 

irom the figures in these tables the curves (Plate I.) were 
plotted. Owing to the stoppage of work at the week-end 
it was unfortunately impossible to obtain a continuous ema- 
nation-curve ; but in some cases the curves are linked up 
according to the knowledge gained from the determined 
portions. 


ANALYSIS OF THE CURVES. 


(1) Let us now consider the emanation-curve in conjunction 
with the barometer-curve. Table VI. opposite gives the results. 
The letter D after a date means a decided change in the 
amount of emanation; and the letter R or S means that rain 
or snow accompanied the barometric change. 

From this table we see that, on the whole, a rise in the 
barometer is accompanied by an increase of the amount of 
emanation, a fall of the barometer is accompanied by a decrease 
in the oot: of emanation, while with the barometer fairly 
steady the issue is doubtful. 


Of a Barometrie rise. 


No Effect. 


Increase of 
Emanation, 


Decrease of 
BE aay 
Tmanation, 


Mar, 3-4 (D) 
ele CD) 
Mar, 22-21 37 226 


Mar. 29-o1 (S) 
Mar. 31-Ap. 2 (D) 


Before May 10 

May 17-19(D) 

May 26-28 (D) 

- Before June 14 
June 1-3 

5» 26-18 Di 


Before May 3 


Aug..18-20 
(R)) Aug. 24-26(DR) 


Of a fairly steady Barometer, or a maximum | 
or minimum. 


| 
mig "2 ee ae 
| 


No Effect. Decrease. Increase. No Effect. 
Mar. 2-4 
|Mar. 17-19(D) 
{ 
May 5-7 May 3-5 | 
| 
| 
May 10-13 (D) | 
» 18-20(D) 
| May 31-dune 2(DR) 
June 7-10 June 9-11 
rae 18) 
Aug, 4-6 Aug. 2-5 (D) 
oe Eke) » 10-12(D)" 
», 23-25 (D): 


Of a Barometric drop. 


in —-j— 


Decrease. Increase. 


Before Mar. 15 (S) 
Mar. 16-18 
, 28-25 (R) 
Apr. 28-29 (R) lo 
(é) 
May 12-14 
May 19-21 (D) 
24-26 (R) 


39 


\ 


June 2-4 (R) 


Aug, 16-18 (R) 


20 Mr. J. Satterly on the Amount of Radium 


(2) It is to be noticed that in England rain nearly always 
accompanies a decrease in the amount of emanation, so that it 
is advisable next to draw upa table showing the effect of rain 
(D again means a decided change in the amount of emanation). 


TasBLe VII.—HEffeet of Rain (or No Rain). 


Decrease Increase 
No Biect. re. Tnerene: when no rain. | when no rain. 
Before Mar. 15 | 
Mar, 2-4(Snow)| Mar. 23-25 (D) || Mar. 15-17 (D)| Mar. 3-5 (D) 
5, 29-Apl. 1 (D) IT ,, dl-Ap. 2(D 
April 27-30 | May 4-7 May 3-5 
| 5, 10-18(D) | ,, 12-14 
May 24-27 | », 1&21(D)] ,, 17-19(D) 
, 3l-Jdune 2 (D) | », 27-28 (D) 
Before Aug. 2 | June 7-10 June 16-18(D) 
| 4, 14-17(D) 
Aug. 16-20 (D) Aug. 4-6 Aug. 2-5 (D) 
» 24-26(D) Aug, 23-25 » 11-18(D)| 4, 10-nzUb) 
(little rain) 


From this table we see that rain (very light falls are not 
considered) is accompanied by a decrease in the amount of 
emanation. There are, however, about an equal number of de- 
creases as well as increases which are not accompanied by rain. 


(3) Coordinated with barometric changes and rainfalJ is 
wind intensity. The next table shows the effect of wind force 
(wind force is not plotted in the curves on Plate I., but is 
given in the tables of results, Tables III., IV., V., pp. 10-17). 
Wind changes of less than two steps on the Beaufort scale are 
not considered. The letter R again means rain (or snow). 


TasLE VIII.—Effect of Wind E orce. 


: 
Of an Increase of Wind Force. '| Of a Decrease in Wind Force. 


None. Decrease. Increase.|| None. |Decrease. Increase, 
Mar. 3-5 
Mar. 16-18 5 16-7 
23-25 (R) 
», 29-Ap. 1 (BR) », dl-Ap. 2) 
April 26-29 (BR) 
May 4-6 
» 11-13 May 13-14 


» 19-20 » 189 
», 20-27 (R) » 26-28 
May 31-June 1 (R) 
Aug. 17-19 (R) Aug, 2-5 


~~ 


ana 


Emanation in the Lower Regions of the Atmosphere. 21 


The results here are even more definite. An increase of 
wind is accompanied by a decrease in the amount of ema- 
uation and a decrease of wind is accompanied by an 
increase of the amount of emanation. 


(4) Considerations of barometric pressure and wind lead 
up to a consideration ef cyclones and articyclones. 
Search was now made in the ‘ Weekly Weather Reports’ 
of the Meteorological Office for cyclones, anticyclones, and 
V-shaped depressions, with the following results :— 


CYCLONES. 
Mar. 1-5. Lazgeirregular shallow cyclone shifting about the NorthSea. 
15-l7a. pees centre in Denmark. 
17-19. Deep cyclone approaching from the west. 
24-26. i, - crossed England ; centre crossed Yorkshire on 


the night of the 24th. 
29-81. Cyclone crossed England ; centre passed Lincoln at noon on 
the 30th. 
May 25-28. Deep cyclone advancing ; centre reached the Irish Sea, and 
then retreated northwards. 
June 21-22, Cyclone appreaching. 
Aug. 17-18. pS Vs ; its centre at Bristol on the morning 
ef the 18th. Then it broke up. 
» 19-12. Big cyclone; centre between Iceland and Norway. 
W winds in England. 
5, 23-26. Cyclone crossed Great Britain, centre crossed Scotland on 
the 24th. 


ANTICYCLONES. 


April 1-2. Anticyclone travelled down from the north and settled with 
its north-south ridge over Great Britain. 
May 3-7. Large irregular anticyclone, centre over Scandinavia. 


fe) wtO-—12. ie protruding over England. 
June 7-11. ye anticyclone advanced, then retreated. 
» 14-18, ee » ; centre off Spain. (This with 


a huge cyclone in Russia gave us cold N winds.) 
Aug. 2-13. Large irregular anticyclone moved eastwards over Central 
Europe; centre in Denmark on Sth. Centre retreated 
to the Atlantic on the 8th, and advanced again ; centre 
in France on the 12th. 


V=SHAPED DEPRESSIONS. 


June 1-8. Large V-shaped depression travelled Eastwards across the 


United Kingdom. 


In all the above cases the cyclones gave rain, strong winds, 
and low emanation values. The anticyclones gave us fine 


weather, and in some cases high emanation values (May 10-12, 
Aug. 3-6, 10-13), and in other cases low emanation values 


(Ma: y 3-7, June 7-11, 14-18). Anticyclones are by no means 


1389 


22 Mr. J. Satterly on the Amount of Radium 


as definite in their structure as cyclones. The only V-shaped 
depression experienced gave plenty of rain, strong winds, and 
a decided decrease in emanation content. | 

(5) The effect of jine weather was analysed ina similar way 
to the above. In 1 case there was no effect on the amount 
of emanation, in 5 cases a decrease of emanation, and in 15 
cases an increase. There were also 3 cases of increase on _ 
dull days (trace of rain only), and 12 cases of decrease on 
days which were not fine (these were chieily rainy days). 

(6) The path of the wind. One would expect that if the 
air that was carried over Cambridge had travelled fora long 
time over land, it would have a large emanation content; 
while if it had travelled over the sea or very rapidly over 
the land, it would have a small emanation content. It is 
not an easy matter to work out the actual path of the surface 
wind. Shaw and Lempfert* have worked it out for certain 
selected cyclones, anticyclones, and V-shaped depressions, 
and have traced the life-history of the currents in great detail 
in their interesting paper. The path of the air has, however, 
to be traced very warily for the conditions are often very 
indefinite. Dr. Shaw was kind enough to let me go up to 
the Meteorological Office and work out the paths of the winds 
from the Working Charts kept there. These working charts 
are synchronous maps giving the distribution of the meteoro- 
logical elements at specified instants. They are got out 
three times a day, 7 a.M., 1 P.M., and 6 P.M., and are used for 
the weather forecastst+. On these charts the isobars are 
drawn for every ;'5 in. of pressure and the wind indicated by 
arrows, the barbs and feathers showing the strength on the 
Beaufort scale, or bya simple calculation in miles per hour. 

The method adopted by me in drawing the surface trajec- 
tories is that given by Shaw and Lempfert in their paper. 
They say: “If we take the synoptic chart for any epoch we 
know to a moderate degree of approximation the speed and 
direction of the wind. The wind observations give us the 
infermation for certain points, and the known relation between 
wind and barometric gradient helps us to interpolate for 
points on the chart for which no actual wind observations 
exist. The continuous records of anemographs show in detail 
what the nature of the changes were for particular localities. 

“A knowledge of the direction of the wind at any point 

* See Shaw and Lempfert, ‘The Life History of Surface Air-Currents. 
A Study of the Surface Trajectories of Moving Air.’ Published by the 
authority of the Meteorological Committee, 1906, 

+ An interesting elementary account of the Construction and Reading 


cf Weather Maps is given by E, Gold in the ‘School World’ of July 
August, and September, 1909. 


Emanation in the Lower Kegions of the Atmosphere. 23 


enables us to draw a step in the surface trajectory which 
passes through the point if we can assume the average move- 
ment of the air to have remained constant during a sufficient 
interval. Thus, for example, if an observation gives the 
wind direction at a station as 8.W. and its speed the equi- 
valent of 20 miles per hour, we may suppose that within the 
half-hour preceding the observation the air travelled 10 miles 
from the South-West, and in the succeeding half-hour it 
travelled 10 miles further towards the North-East. For 
longer periods a proportionately longer step must be drawn. 
mee S . So long as the motion of the air is of considerable 
magnitude and remains persistent for a considerable time, 
there is little difficulty in drawing the steps with considerable 
confidence; but when we have to deal with a region of light 
airs, and in the outlying region of an approaching depression, 
or when the changes are rapid, as in the region of the centre 
of a depression, the drawi ing of the trajectory is an uncertain 
process.’ 

Shaw and Lempfert’s air trajectories over the United 
Kingdom were obtained from charts got out every hour or 
two hours. ‘The charts at my disposal had successive intervals 
of 6, 5, 13 hours, so that the trajectories given in Plate II. 
are not very trustworthy. Still in the absence of anything 
better they serve a useful purpose. 

Again my ‘‘exposures”’ lasted 21 honrs, so that the trajec- 
tory of the air arriving at the beginning of the exposure 
might be very different from the trajectory of the air arriving 
at the end of the exposure. To get over this difficulty iT 
have drawn in some cases trajectories at successive intervals 
of about a day. ‘The course of the trajectories of the air 
arriving at Cambridge between the instants taken may thus 
be roughly approximated to by a consideration of the known 
trajectories. 


Trajectories of the surface air-currents arriving at Cambridge 
when a low value of the emanation content was obtained 


(Maps 1, 2). 


Curves (1), (2), (8) of Map 1 give the three trajectories for 
the surface air arriving at Cambridge on March 24, 6 P.M., 
March 25, 7 a.M., March 25, 6 p.m., during the passage of a 
deep cyclone, the path of whose centre is represented by the 
dotted line. The velocity of the centre was 20 miles per 
hour. The wind velocity was from 5 to 7 on the Beaufort 
scale, 2.¢. from 15 to 35 miles per hour, so that the cyclone 
would be said to be a fast traveller. ‘The air represented by 
curve (2) covered its 320 miles of land travel in 20 hours, 


24 Mr. J. Satterly on the Amount of Radium 


so that it had not much time for picking up emanation from 
the land over which it passed. 

Carve (4) of Map 1 gives the trajectory for the air arriving 
at Cambridge on June 16 at 6 p.m. It was traced back to 
June 13, when it was found to come from a northern region 
of gentle airs between an Icelandic high pressure region and 
a Norwegian low pressure region. The air leisurely pursued 
its southerly course to Cambridge, being over the sea all the 
time except for the last 50 miles. Its emanation-content 
would naturally be low. 

Curves (1) and (2) of Map 2 give the trajectories for the 
air arriving at Cambridge on June 2, 6 P.m., and June 3, 
6 pM. The air had travelled at a great pace across the 
Northern Atlantic between a cyclone in the Icelandic region 
and an anticylone to the south of its course. It hung about 
tor a short time in a quiescent region around the Shetlands, 
and finally travelled southwards at a great pace behind a 
very long V-shaped depression which was travelling east- 
wards to Denmark, and which gave us heavy rains on the 
evening of June 1 and the morning of June 2. 

Curve (3) of Map 2 gives the trajectory of the air arriving 
at Cambridge on May 25,6 p.m. The air was traced back 
to the quiet interior of an anticyclone which had hung about 
over the continent since May 17. No doubt the interior of 
this anticyclone was occupied by air which had just descended 
from the upper regions so that the surface trajectory could 
not be traced beyond the 24th. The air reaching Cambridge 
had thus made little contact with land, thus explaining its 
low emanation content. 

All the paths of the trajectories mentioned above agree 
very well with the small amount of emanation the air in 
them carried. 

The trajectories for the air giving high results do not give 
such good agreement. This is no doubt partly due to the 
fact that high results occur in fine weather and the winds 
in fine weather are light and variable. 


Trajectories of the surface air-currents arriving at Cambridge 
when a high value of the emanation content was obtained 


(Maps 3, 4, 5). 
Curve (1) of Map 3 gives the trajectory of the air arriving 


at Cambridge on March 5,7 a.m. There had been a number 
_of .sporadic cyclones in N.W. Europe for the past 4 days, 


and the winds had. been light. The air. was .traced back 


Emanation in the Lower Regions of the Atmosphere. 25 


through the English Midlands and Yorkshire to Norway, 
where it probably descended in the anticyclone situated 
there on Feb. 27. 

Curves (2) and (3) of Map 3 give the possible trajectories 
of the air arriving at Cambridge on Aug. 5, 7 a.m. The 
wind at Cambridge was just S.of W. If on drawing the 
trajectory a WSW. wind is taken, the path leads back by 
curve (2) to a point on an anticyclonic ridge to the south of 
Cornwall on Aug. 3, and it cannot be traced further. If 
the wind is taken due W. the path leads back hy curve (3) 
to S. Wales, and then by strong northerly winds to Scotland. 
It is quite doubtful from the working charts which path 
should be taken, but the emanation content would incline 
one to the path from Scotland, although the existence of 
uranium-radium-mines in Cornwall must not be forgotten. 

Curves (1), (2), and (3) of Map 4 give the trajectories of 
the air arriving at Cambridge on May 10, 6 p.m, May 11, 
6 p.m., May 12, 6 pm. All the air had come from the 
northern region, ‘and from the paths of the trajectories one 
would expect at first sight (1) to give a low emanation- 
content, (2) a larger content, and (3) larger still; the actual 
results were just the reverse (see Table IV. A). The weather 
phenomena over England were, however, anticyclonic on May 
10th and cyclonic with freshening winds on May 11th and 
12th. 

Map 5 gives ites possible trajectories for the air arriving 
at Cambridge on May 19, 7 a.m. The observers for the 
Meteorological Office estimate the wind force on the Beaufort 
scale, and their estimate is of its nature only very approximate. 
A wind estimated as No. 3 on this scale may be anything 
between 8 and 13 miles per hour with a mean value of 10 
miles per hour. A wind estimated as No.4 on the scale 
may be anything between 13 and 18 miles per hour with a 
mean of 15 miles per hour. It will be observed that the 
minimum wind velocity according to one number on the 
scale is the maximum wind velocity allowable for the next 
lower scale-number. Thus there is a fair latitude allowed 
for different observers; and probably some will always over- 
estimate and some always underestimate. On Map 5 I have 
plotted three trajectories corresponding to the maximum, 
mean, and minimum values of wind force corresponding to 
the numbers of the Beaufort scale. The pressure conditions 
were anticyclonic on the continent on May 18 and 19, feebly 
cyclonic over Britain, France, and Spain on May 16 and £7, 
‘and there was a persistent calm. to the west of Ireland on 
“May 15 and 16, with prevailing northerly winds from Iceland 


26 Mr. J. Satterly on the Amount of Radium 


to west of our islands previous to this. The curves corre- 
sponding to the maximum and mean wind values agree very 
well, both going through the calm to the west of Ireland. 
The minimum curve did not reach this calm, but was caught 
in a feeble cyclone between Wales and Ireland, where it 
performed a loop. The winds were gentle, spent a long time 
over land, and reached Cambridge in fine warm weather, 
thus accounting for a large emanation content. 


The average amount of Radium Emanation in the Arr. 


In order to express the quantity of radium emanation in 
the air in terms of the quantity of radium which would be 
in radioactive equilibrium with it, it is necessary to carry 
out comparison experiments, placing a radium solution of 
known strength in series with one of the tubes. The arrange- 
ment of fig. 2 is modified thus :— 


Y daisy A—>Gauge A— 
Outside air — — Pump, 


\ Rad. Sol.»+Tube B>+Gauge B> 


The radium solution used was kindly given to me by 
Prof. Rutherford, and its radium content was 3°14 x 10-® om. 
. . ’ 99 = 
in the form of bromide. Before any “ exposure” the solution 
was cleared of all accumulation by bubbling air through it 
for an hour or two. The same volume of air is sent along 
each path at the same rate, therefore from the amounts of 
emanation found from the two tubes we can get by subtraction 
the value of the ratio 


emanation in a known volume of air 
emanation generated by the solution in a known time. 


The method is a comparative one, and the results are only 
true if the same fraction of the total emanation is absorbed 
by the charcoal, whatever be the emanation-content of the 
air sent through the charcoal-tubes. Jixperiments made with 
strong solutions indicated that with strong solutions satura- 
tion occcurred, but with solutions giving about the same 
amounts of emanation as those obtained in the experiments 
fairly satisfactory results were obtained. 

The denominator in the fraction given above is the amount 
of emanation yielded by asolution containing 3°14 x 10-° gm. 
radium in the known time, ¢, of the experiment. To find 
the mass of radium which would be in radioactive equilibrium 


Emanation in the Lower Regions of the Atmosphere. 27 


with this amount of emanation we multiply the denominator 
hy 
EAL, 
1, ie Ty (1 ih) or J—e*, 
using the same notation as before (p. 6). 
The following table gives the values of 1—e~** required 
in the experiments :— 


TA Bie) Lk. 


} 


] | ¢ (hours) .... 3. Ze let Pee a oD. 20 
| wet Ry cb c's I a einen | pee 
a | 
Bt 9 0228 | -O5LL) |} 0907) 146 152 1-000 
| 
; Thus the amount of radium that would he in radioactive 
; equilibrium with the amount of emanation that my solution 


generates in 3 hours is 
3°14 x 10-* x :0223 om. 


Five runs of 3 hours each were made with the tubes in 
parallel, the solution being in series with one of the tubes. 
The results are given in the following table :— 


TABLE X.—Short Runs. 


| 7 
| | 


Air alone. | Air plus Solution. | Emanation 
ee eee ay a | generated 
ie Be Vol. of [ae Rages Vol. of | ReneS eet eis 
Tube. fits, air. | caught. ae ee | air. | caught. bee beg 
re ee | 10 B | 58 | 106 | a eae 
See Meo Ge Nhs Bylo r® comtetends ss Bao. £9 
But jh, 908 Ie. | 5 B | 5 | G0 (4 51 =| 46 | 
D | Se ie |B) aL aa 58 by 2a 
| AY | eens Bon iad ui 44 


The average amount of emanation caught from the air in 
3 hours is *8, and the average amount caught from tlie 
solution (alone) in 3 hours is 4°65 (neglecting the reading 
on Mar. 2 when the speed of the air-stream rose to °58). 

Therefore on the average the emanation in 90 litres of 
atmospheric air is equal to that which would be in radioactive 
equilibrium with | 
: x 3°14 K 10>? x ‘0223 gm. of radium. 


poe 
Or 


28 Mr. J. Satterly on the Amount of Radium 


The amount of radium which would be in radioactive 
equilibrium with the emanation in one cubic metre of air is 
called by Eve the Radium Equivalent. Therefore the average 
radium equivalent on the days given in Table X. is 


1000 My 8 
JO 4°65 
= 130 x 10-” om.* 


In the 1908 experiments the porcelain tubes absorbed in 
two hours an amount 2°5 from the radium solution. The 
silica tubes with more than 4 times the cross-section of the 
porcelain tubes only absorb 4°65 in three hours. This looks as 
if the charcoal tended to get saturated with the emanation, 
so that the amount absorbed is not proportional to the time 
of exposure. ‘The air-readings also tend to give the same 
conclusion for the average value of the emanation caught 
from the air in 21 hours (deduced from the results of March, 
April, May, June, and August) is 2°45, which is much less 
than 7 times *8 the average value of the air caught in 3 
hours. 

These facts suggest that the absorption of the emanation by 
the charcoal is two-fold, one stage being quick—a surface 
condensation, and the other much longer—a diffusion into 
the interior, as has been discovered by McBaint for the 
absorption of hydrogen by charcoal. 

These experiments have led to other experiments on 
saturation which will be published in due course. 

In order to get solution readings to compare with the 
21-hours air readings, it is therefore necessary to make 
solution-exposures of 21 hours. The radium solution used 
above is too strong for this purpose, as it is wise:to keep the 
amount of emanation from the solution about equal to that 
from the air, soa solution of one-fifth strength, 7. e. containing 
3°14 x 10-9 

5 


the results of the experiments given in the following table 
allowance has been made for the fact that the absorptive 
power of tube B was about 10 per cent. better than that of 
tube A. (This is shown by the air-results of June and 
August.) The air-stream was kept practically constant at 
*5 litre per minute. 


x B14 x 10-8 x 0223 


gm. of radium, was made up. In calculating 


* The calculation is not quite correct as no allowance is made for the 
decay of the first portions of the emanation taken from the air, whereas 
the formula used for the solution allows for this. The error, however, 
is practically negligible. .- 

+ Phil. Mag. Dec. 1909. 


Emanation in the Lower Regions of the Atmosphere. 29 


Taste XI.—Long Runs. 


Dura- Air alone. | Air plus Solution. 


: | 
| 

Date tion of Emanation | Radium | 

a expo- eerie SBunaiae caught from equivalent | 

eo? -- Soe Tube. | Vol. P| tion || Tube. Vol. oF tion | the Solution. be the Air. | 
air. | caught. | alr. | a ught. | | 
Rei eho mm ke oa 
Oct. 26-27) 222 | A | 630/ 11 || B | 660) 47 | 47—1:2=35 | 52x10 | 
» 27-98| 21 | B | 610 t eee A 640) 41 | 41— 8=33| 87x10 
| | 9] 

Nov.2-8.... 21 | B | 650/ 22 | A | 650 | 52 | 52-20=32 | 98x107") 
| | | —12! 
ee a a | on 26 B | 635 67 | 67-29=38 |109x10 ~ 


} { 
i } 


To calculate the mean emanation content of the air for the 

“months March—August we use the mean values 2°45 for the 

average value of the emanation from 680 litres of air, and 

3°45 for the amount of emanation absorbed from a lass 

tex 10-? 
5, 


Therefore the average radium-equivalent of the air 


ee ae, d*14x 10-9 
Dy 650) 24. D 


= 105 x 10-™ on. 


The lowest value is about 35 x10-” om. and the highest 
value about 350 x10-” gm. 


containing om. of radium in 21 hours. 


x 146 


Discussion of the results of experiments made on the variation 
of the amount of radium emanation in the air with weather 
conditions. 


The only other experiments besides the author’s on the 
measurement of the amount of radium emanation in the air 
S and its association with meteorological phenomena are those 
| of Eve* made at Montreal. The method was the same, but 
_ Hve’s exposures lasted 2°7 days, whereas mine lasted only 
21 hours. Eve found that the ratio of the greatest to the 
least values was 7:1. I get 10:1. He found also that 
the amounts in summer and winter were not widely different; 

this also agrees with my results. 
: Hve also found that the approach of a deep cyclone, ac- 
a companied by heavy rain or a quick fall of snow, causes an 


; * Phil, Mag. Oct. 1908. 


tite =r enespnresseensrenant : a —— —— = = = = a2 Se 


ee 


30 Mr. J. Satterly on the Amount of Radium 


increase in the amount of radium emanation in the air, whilst 
anticyclonic conditions, with dry or very cold weather, give 
a decrease in the amount of the emanation in the air. This 
he explains 


(1) by the spiral motions of cyclones and anticyclones, 

(2) by the suction action of changes of pressure on the 
emanation in the air lodged in the ground; 

(3) by the readier liberation of emanation from moist 
substances. 


My results are opposed to Eve’s in this respect. I find 
that cyclones accompanied by wet weather and strong winds 
give a decrease in the amount of emanation, whilst anti- 
cyclones accompanied by fine weather and light winds give 
an increase in the amount. There are exceptions, but the 
above is generally true, and the trajectories I have drawn 
bear it out. 

Of course there is a great difference in the situations of 
Montreal and Cambridge. The cyclones crossing England 
come straight from the Atlantic, while those arriving at 
Montreal have travelled over hundreds of miles of lund, and 
this may explain the whole difference. If my experiments, 
however, had each lasted 3 days, I should have found much 
less difference between the maxima and minima values. 
Ashman *, working at Chicago with the liquid air method, 
made six measurements of the emanation content, and he 
gets practically the same results as Eve. 

Most of the other workers in this subject have exposed 
charged wires to the air, and measured the active deposits 
obtained, the others have measured the atmospheric ioniza- 
tion, &e. Dyke ft, working at Cambridge for three weeks in 
1906, drew a measured volume of air through a negatively 
charged metal grid for an hour at a time, and found a 
greater deposit on still bright days than on cloudy windy 
days, and a small amount after rain. In California Harveyf 
found the largest deposits occurred when a land wind blew 
and the humidity was low, and the smallest deposits when 
an ocean wind blew and the humidity was high. Work has 
been done on the Continent of Europe by Gockel §, Kohl- 
rausch||, Schweidler{], Amaduzzi**, Constanzotf, Simpsonff, 


* Amer. Journ. Sci. Aug. 1908. 

+ Terr. Mag. and Atmos. Elec., Sept. 1906. 

t Le Radium, Aug. 1909. 

§ Phys. Zeitschr. May 1908. 

|| Akad. Wiss. Wren, Sitz.-Ber. Oct 1906. «| Ibid. 
** Accad. Lincei, Atti, Jan. 1909. 
tt Phys. Zeitschr. Mar. 1909, 
ft Phil. Trans. 1905. 


Emanation in the Lower Regions of the Atmosphere. 31 


Flemming™, and others. Gockel confirmed the conclusions 
of Brandes, namely, when the barometer is low there is more 
active deposit on a wire, and says that rain and gusty weather 
produce an increase in the amount of emanation. In an 
earlier paper he says there was more active deposit on fine days 
than on wet ones. Kohlrausch, working in Vienna on the 
radioactive induction, found that clouds had a considerable 
effect, and that there was a decrease after rain and with 
falling pressure; whereas Schweidler found an increase in the 
atmospheric ionization on stormy days. Amaduzzi supports 
Ebert’s: contention that a fall of the barometer causes an 
increase in the amount of emanation in the air. Constanzo 
measured the active deposit over the Mediterranean and 
found it increased when the wind blew from the land. 
Some of the best work has been done by Simpson, who 
carried out a long series of experiments in Lapland in 1905 
on the active deposit from atmospheric radioactivity. He 
found that 


(1) On the whole year temperature has a marked effect, 
but little effect during any one month; 

(2) The radioactivity increases as the humidity increases 
and decreases as the humidity decreases ; 

(3) The radioactivity decreases as the wind increases in 
strength ; 

(4) The radioactivity is greater with a falling barometer 
than with a rising barometer, but the radioactivity 
is not necessarily higher wae a low barometer than 
with a high one; 

(5) The radioactivity is greater with winds from the land 
than with winds from the sea. (In Lapland the 
winds from the land are South winds, and occur when 
the barometer falls. The winds from the sea are 
North winds, and occur when the barometer rises.) 
The wind effect was due to the barometer effect and 
not vice versa. 

(6) There is no connexion with the amount of cloud and 
the radioactivity. 

(7) There is no relation between the radioactivity and the 
potential gradient. 


Simpson concluded by saying that all his work supports 
Elster and Geitel’s view that when the atmospheric pressure 
falls the emanation is sucked out of the ground. He also 
found that everything which reduces the atmospheric circu- 
lation increases the atmospheric radioactivity. 


* Phys. Zeitschr. Nov. 1908. 


3B Mr. J. Satterly on the Amount of Radium 

Workers on the ionization in closed vessels include Campbell 
and Wood* at Cambridge, Wulff at Valkenberg in Holland, 
Strong ¢ at Baltimore, Pacini§ at Sestola, Wright || and 
Cline {] at Toronto, and many others. Nearly all European 
observers find a double daily variation in the ionization which 
is closely parallel to the changes in the atmospheric potential 
gradient. Wright (loc. cit.) and Cline, however, find no 
evidence of a regular daily variation. Campbell and Wood, 
Pacini and Cline agree in finding that the ionization is 
independent of the pressure, temperature, and humidity of 
the air. When his vessel was not sealed Cline found that 
the ionization was greatest when the atmospheric pressure 
was lowest, and this he put down to the emanation sucked 
out of the earth. 

The above results show that at present the subject is in a 
confused state. It is desirable that instead of isolated 
observers working on different points of the subject, well 
equipped bands of observers at several laboratories should 
thresh out the subject properly. Evidently much depends 
on the locality at which the observations are made. Hxposed 
wires do not seem to lend themselves to great accuracy in 
the measurement of atmospheric radioactivity, as there is 
little knowledge of the actual volume of air which has con- 
tributed to the deposit on the wire. Dyke’s method seems 
to be the best for measuring the active deposit, and the 
charcoal absorption method for measuring the amount of 
emanation in the air. 


Amount of Ionization of the Air due to the Radium 
Emanation present. 


In my former paper I calculated that the radium emanation 
in the air was, on the average, responsible for the formation 
of 2°} ions per c.c. per sec. Hve **, using later data for the 
number of a particles shot out per sec. per gm. of radium 
and its products, reduced this to 1°3 ions per c.c. per sec. 
Recently Geiger ff has determined afresh the average number 
of ions produced by an a particle in its flight. He finds 
that the average number of ions produced by an @ particle 


* Phil. Mag. Feb. 1907. + Phys. Zettschr. Mar. 1909. 
{ Phys. Review, Feb. & July 1908. § Accad. Lincei, Atti, Feb. 1909. 
|| Phil. Mag. Feb. 1909. “| Phys. Review, Jan. 1910. 


** Kve, ‘ Terrestrial Magnetism,’ March 1909. 
tt Geiger, Roy. Soc. Proc. vol. Ixxxii., July 1909. 


Emanation in the Lower Reaions of the Atmosphere. 33 


from radium emanation is 1°74x10°, from Radium A 
1°87 x 10°, and from Radium C 2:37 x10. The average for 
these three products is 2°0 x 10°. Hence working on the same 
lines as before we find that the number of ions produced 
per sec. per cub. metre of air by the radium emanation, 
radium A, and radium OC in radioactive equilibrium with 
105 x 10—” grm. of radium is 


(105 x 10-”) x3 x (34x 10") x (2:0 x 10°) 
Gea a LOe, 
or about 2 per c.c. per sec. 


When the emanation content is at its lowest value this 
would be reduced to ‘7, and when at its highest value would 
be raised to 7. | 

The average emanation content at Montreal, as measured 
by Eve, is 38; of the amount in Cambridge, so that the 
number of ions produced by the emanation at Montreal per 
c.c. per sec. is about 1:2. 


Amount of Ionization of the Air due to the Thorium 
Emanation present. 


W. Wilson* has shown that there is about 4000 times as 
much radium emanation as thorium emanation in the atmo- 
sphere near the earth’s surface. The radioactive constant of 
thorium emanation is about 5000 times greater that of 
radium emanation, hence there are about the same number 
of thorium emanation atoms and radium emanation breaking 
up per second in the atmosphere near the earth, and there- 
fore the thorium series of disinteyrating products would also 
be responsible tor about 2 ions per c.c. per second. 

Therefore in free air the number of ions produced per c.c. 
per sec. by the emanation of radium and thorium present is 
about 4. If a vessel were slowly filled with filtered air the 
thorium products would not enter the vessel, and therefore 
only 2 of the number of ions produced per c.c. per sec. in a 
closed vessel are due to emanations, and those 2 to radium 
emanation. 


Apportionment of the .ons produced in a metal vessel and in 
free air to the ionizing agents at work. 


The number of ions produced per c.c. per second in a metal 
vessel has rarely been reduced below 10. This number has 


* Phil. Mag. Feb. 1909. 
Plul. Mag. Ser. 6. Vol. 20. No. 115. July 1910. D 


d+ Mr. J. Satterly on the Amount of Radium 


been reached by Cooke*, who got 9:1 in a brass vessel 
screened by large masses of lead, “and by Wright ft, who got 
down to 6:0 in an unscreened zinc vessel on the surface of 
Lake Ontario, and to 8°6 at the same time in a similar lead 
vessel. In the calculation of their results all observers 
have taken the old and small value of e, the electronic charge. 
Wright took e=3'4x10-' us.u. If we substitute the 
correct value e=4°65x10-" zE.s.u., Wright’s figures for 
the number of ions become 4°4 and 6°3 respectively. Wright 
also found 8:2 and 9°8 for the same two vessels in the open air 
on the ground near the newly erected Physical laboratory at 
Toronto, and 9°9 and 10:3 in a room within the laboratory. 
In each case the vessel had been cleaned and filled with 
filtered air just before the readings were taken, and was 
unscreened. 

McLennan has shown that the water of Lake Ontario is 
quite free from radioactive matter {, and Wright has shown 
that if the depth of water is greater than four metres it 
absorbs all the penetrating radiation from the earth below 
the lake. Hence it follows, considering the zinc vessel alone, 
that the diminution from 8:2 to 4-4 was due to the cutting 
off of the penetrating radiation from the earth. Allowing 
1 ion per c.c. per sec. to be due to the radium emanation in 
the air in the vessel, it follows that the remainder must be 
due to an intrinsic radiation from the walls of the zinc vessel 
itself. Of the total number 8 produced on the ground near 
the laboratory, we may therefore say that (a) 4 are due to 
the penetrating radiation from the ground (including any 
secondary effect this may produce), (b) 3 are due to a radia- 
tion from the walls of the zinc vessel itself, and (c) 1 is due 
to the radium emanation in the air. In free air (a) would 
be reduced to 1 (see Wright, loc. cit. p. 317), (b) would be 
absent, and (c}) would be raised to 2 by the presence of the 
thorium products, giving a total of about 3 ions per c.e. 
per sec. in free air. 


Volume of Radium Emanation in the Avr. 


Rutherford§ has shown that the volume of radium emana- 
tion in equilibrium with 1 gm. of radium is 585 eub. mm. 


* Phil. Mag. Oct. 1903. + Phil. Mag. Feb. 1909. 

{ Eve (Phil. Mag. July 1909) has shown that the radium content of 
the St. Lawrence at Montreal is -25 10-12 gm. of radium per litre of 
the water. 

§ Proc. Roy. Soc., Aug. 1908. 


Emanation in the Lower Regions of the Atmosphere. 35 


Hence the average volume of radium emanation in one 
cubic metre of the atmosphere at Cambridge 


==( L0a «105 x (d80. x LO.) 
=; aw On ae.c. 


or there is 6°1x10-” c.c. of radium emanation in | c.c. of 
air. 

Now there are 2°76 x 10° molecules of gas in 1 c.c. of gas 
at 0° C. and 760 mm. Therefore the average number of 
emanation molecules in 1 c.c. of the atmosphere near the 
earth’s surface 

=O e105 ie (276 10P) 
il Il 

Surely it is a triumph to be able to detect the existence of 
a gas in the atmosphere when there are less than 2 molecules 
of it present in a cubic centimetre. 


Volume of Helium produced from the Emanation in the Air. 


Rutherford and Boltwood* have shown that 1 gm. of 
radium produces helium at the rate of 163 cub. mm. per year. 
Therefore the amount of helium produced per year from the 
radium emanation and its products in 1 c.c. of the atmosphere 


=2x-°16x(105x10-"x 10-°) 
=1:3x10-" «ea, 


Allowing an equal amount to be produced from the thorium 
products in the atmosphere, and realizing that most of the 
helium produced by the disintegration of the earlier members 
of the series in the earth’s crust is kept there, it follows that 
3x 10-1!" ¢.c. of helium is produced by radioactive processes 
per c.c. of the atmosphere per year. 

The amount of helium actually present in 1 c.c. of the 
atmosphere is about 5 x 10-® ¢.c., so that these figures hardly 
afford a method of calculating the age of the earth. 


SUMMARY. 


(1) The amount of radium emanation in the atmosphere 
near the earth’s surface at Cambridge has been measured at 
intervals during a year by the coconut charcoal method. 

2) The average radium equivalent per cubic metre is 
105x10- gm. The lowest value is 35x 10-” gm.,and the 


highest 350 x 10- om., a ratio of 1: 10. 


(3) The amount of emanation is usually lowest during 
* Manchester Lit. and Phil. Sor. Mem. 54, 1909-1910, 
D2 


gana 
S——_- 


SSS SES 


eee —-—- = =—=_- = ——— —— ee a SS SS 2 en eS , an oe | ee Se Se ee Saye a 
—S eS = er > — SSE —— = = —- ——— ——— —— =—— ———= — 


36 Mr. A. L. Fletcher on the Radioactivity of 


cyclones, 2. e. during windy, wet weather, when the barometer 
is low; and usually highest during anticyclones, i. e. during 
dry weather with light variable winds ani a high barometer. 

(4) In cases where (3) breaks down, a study of the tra- 
jectories of the surface air-currents reveals that when air has 
travelled over the sea to Cambridge or very rapidly over 
land, the emanation-content is low, while if the air has spent 
much time over land, the emanation-content is high. 

(5) The results of other experiments have been discussed. 
Eve’s results obtained by a similar method at Montreal difter 
from the author’s, but this is probably due to different geo- 
graphical conditions. Experiments made by measuring the 
active deposits on exposed wires seem to give misleading 
and indefinite results. 

(6) The number of ions produced per c.c. per sec. in free 
air at Cambridge due to the radium emanation present is 
about 2°1 on the average, with a minimum value of ‘7 and a 
maximum value of 7:0. 


In conclusion the author wishes to thank Prof. Sir J. J. 
Thomson for permission to carry out these experiments in 
the (‘avendish Laboratory, and for his stimulating suggestions. 
The author also wishes to express his thanks to Dr. W. N. 
Shaw and his assistants at the Meteorological Office for help 
in tracing the trajectories of the air. 

Cavendish Laboratory, Cambridge, 

April 1910. 


a’. 


| II. On the Pad abaiwelh ope the Rocks ” the Transandine 


oN Tunnel. By Arnoup L. Firrcuer, B.A.1.* 


are following determinations of the Radium content of a 


series of volcanic rocks from the Andes of South 
America were made as the result of a suggestion by Pro- 
fessor Joly, that a systematic analysis of some of the larger 
igneous masses was desirable. 

This is the more apparent in view of the various results 
obtained upon igneous rocks from different localities, and 
the consequent necessity for a larger number of systematic 
determinations, with a view to the establishment of some sort 
of a mean in the case of such materials. 

The determinations were made upon a series of rock- 
specimens tuken at various points during the working of the 
tunnel, and which were obtained by the kindness of Mr. E. 
Manisty, M.Inst.C.E. They were taken at an average depth 
of ahout 1000 feet below surface-level. 


* Communicated by the Author. 


the Rocks of the Transandine Tunnel. | 37 


Preparation of Solutions. 


Owing to the extreme sensitiveness of the electroscopes 
used, and the small quantity of material dealt with at each 
experiment, every possible precaution was taken throughout 
to avoid the possibility of any errors due to contamination. 
The work was carried out in a room which had never con- 
tained radioactive preparations of any sort. The apparatus 
was—with few safe exceptions—new, and hitherto unused. 

Extreme care was taken to avoid the introduction of radium 
from the use of impure chemicals, it having been observed in 
previous work that the commercial alkalies and hydrochloric 
acid may contain a sufficient quantity of radium to exert a 
noticeable effect upon the electroscopes. Both the water and 
the hydrochloric acid used were distilled in the laboratory. 
In the latter case, the distillation was performed over com- 
mercially pure chloride of sodium, so that 100 c.c. of purified 
acid, evaporated down, showed no trace of sulphuric acid with 
barium chloride; certain experiments having shown that traces 
of sulphuric acid are capable of diminishing the emanating 
radioactive power of rock solutions. 

Care was taken in the preparation of the specimens for 
chemical treatment that they were not exposed to any risks 
attendant upon handling, but were in nearly all cases mani- 
pulated by forceps. The quantities of chemicais used, 
together with a full description of the electroscopes and of 
the method employed, may be found in ‘ Radioactivity and 
Geology,’ chap. xii., and need not therefore be further 
described. 

Jonsiderable difficulty was experienced in obtaining 
solutions free from precipitate ; often three, and sometimes 
even four, refusions having been made. It was, however, 
frequently found that a repetition of the fusion failed to 
render soluble the original precipitate. It may be noted that 
those solutions which contained precipitate, and which in the 
table given below are distinguished with a letter p, appear to 
show no faliing off from the general mean. 


Calibration of the Electroscopes. 


Particular attention was paid to the calibration of the 
electroscopes. Of these, two were continually in use, “A” 
being calibrated to an alkaline, and ‘‘ B” to an acid solution. 
Both were of about 620 c.c. capacity, and much alike in the 
dimensions of the gold-leaf system. The earlier calibrations 
were effected by an observation of the rate of collapse of 


38 Mr. A. L. Fletcher on the Radioactivity of 


the leaf, consequent upon the rapid introduction of the 
emanation, from an aqueous solution containing 1 m.g. 
uraninite. Subsequently the calibration was effected under 
conditions more nearly approximating to those obtaining 
when a rock of low radium content is being examined. To 
this end a quantity of radium was used, more nearly com- 
parable with the amount found in the solutions examined, 
and one of the electroscopes was calibrated from an alkaline 
rock solution, whose radium content had been previously 
determined ; the other similarly from an acid solution. 

The following results will show that the calibration of 
the electroscopes under the new conditions resulted in a 
marked modification in the constant. The following is a 
list of the calibrations. 


ELECTROSCOPE A. 


(1) Standardized from 1 m.g. uraninite (from a standard 
solution of a uraninite containing 64 per cent. uranium, made 
by dissolving the mineral in HNOzg, and diluting to a strength 
of 1 m.g.in 1¢.c.) in about 500 c.c. distilled water ; enclosed 
for 19 hours 25 minutes, involving the accumulation of 
14 per cent. of the equilibrium amount of the emanation. 

The transference was through a capillary tube and occupied 
about ten minutes. 


Gain = 60 scale-divisions per hour. 
From this C= 0°52x10-", 


where C represents the quantity of radium in grams re- 
sponsible for a gain of one scale-division per hour, 
(2) An earlier experiment of the same character gave 


C= 091x107 


In these experiments the emanation in the electroscope 
would be equivalent to an amount in equilibrium with 
31°2x10- gram of radium, i. e. a quantity almost four 
times in excess of the average measured in the experiments, 
in which the gain was seldom over 20 scale-divisions per 
hour. 

(3) Standardized with ‘4 m.g. uraninite in 600 c.c. distilled 
water. Closed 20 hours, 2. e. 14:4 per cent. emanation col- 
lected. Boiled 26 minutes with talc, using slow admission 
to electroscope. HEmanation present in electroscope equivalent 
to 12°5x 10-” gram radium in solution. 


Gain = 24:5 seale-divisions per hour, 
cqavence i 4 Am 51x 10-2”. mah f 


the Rocks of the Transandine Tunnel. 39 


(4) Standardized from the alkaline solution derived from 
10 grams Keuper Sandstone, fused with 24 grams mixed 
carbonates, in 600 c.c. distilled water ; leached and filtered 
clear, which when tested gave a gain of 4 scale-divisions 
per hour. To this was added 2 m.g. uraninite, standard 
solution. Closure 14 days, 2.e, 92°7 per cent. emanation 
collected. 


‘Gain = 65'5 scale-divisions per hour. 
True gain = 70°7 a 3 
if closed till radioactive equilibrium was established. 
Hence 
Gain due to added radium = 66:7 scale-divisions per hour, 


whence = “6h x<105™ 


The constant will be somewhat higher if we assume the 
normal leak the same for both experiments, when 


2nd leak— 1st leak = gain for added radium. 
This comes out as 62 scale-divisions per hour, and 
Cr == "0D x 10s 


(5) Standardized from an alkaline solution obtained by 
fusing 20 grams basalt in 120 grams carbonates, dissolving 
in distilled water, leaching and filtering. 

This was found to contain 3x 10-2 gram radium by 
preliminary experiments, using the constant 0°6 x 10-”. 

To this was added *2 m.g. uraninite from standard solution, 
so that the total radium present was 


43°5 x 10-" + 3x10-¥ gram = 46°5 x 10-¥ gram. 
Closure 20 days. Gain 63°7 scale-divisions per hour. 


wo: 46 -5De 10. 
Hence McG gill | 


If again we calculated by the difference in gain of rate of 
discharge in the 1st experiment (5 scale-divisions per hour) 
and in the 2nd experiment (63°7 scale-divisions per hour), we 
get the 


=)" bo 10, 


Gain due to radium = 58°7 scale-divisions per hour. 


. —12 
ec he pees O10 2 oa 10-1, 
587 


40 Mr. A. L. Fletcher on the Radioactivity of 


ELECTROSCOPE B. 


In the case of this electroscope the tests were made 
throughout with strongly acid solutions. 

(1) In this case the test was carried out with the tem- 
porary addition of a U tube of CaCl, and KHO, and 
with the delivery tube reaching to the bottom of the 
electroscope. 

Standardized with 1 m.g. uraninite in 60 c.c. strong radium- 
free HCl and 600 c.c. distilled water. Closure 44 hours 
30 minutes, 7. ¢. 30 per cent. emanation collected. 


Gain = 82:4 scale-divisions per hour. 


Here the emanation present was equivalent to a quantity 
in equilibrium with an amount of radium = 65°28 x 10-¥ 
gram. 


FA -YR —12 
Hoxve hee 65 28x 10 


arabs pe Sea 9s —12 
Sa = 0°79 x 10-™, 


(2) Standardized from an acid solution of 10 grams Keuper 
Sandstone, consisting of the precipitate insoluble in water 
after fusion with 24 grams fusion mixture, dissolved in 
70 c.c. distilled HCl. Solution very limpid. 

Zest I. Closure 14 days. 


Total leak = 22 scale-divisions per hour 


and gain = 16 oe a 
Test II. (after adding *2 m.g. uraninite from standard solu- 


tion). Closure 14 days. Gain=65°5 scale-divisious per hour. 


Gain due to radium=65'5 —16=49°5 scale-divisions per hour. 


Hence Be i ee ae _19 
C= 195 = Secor a, 

In this case the emanation present represented 40°47 x 10-¥ 
gram radium. 

(3) Standardized from the acid solution of No. 9 of the 
Andes rocks. This had already been found to contain 
9x10- gram radium, using the constant *3. To this was 
added from a standard uraninite solution 8'84x10- gram 
radium. 

Hence 


total radium = 17°84 x 10- gram. 
Gain (closure 21 days) = 25 scale-divisions per hour. 


- -12 
Hence @n2 17°84x 10 Ae 40-1) 


20 


the Rocks of the Transandine Tunnel. Al 


(4) Standardized from the acid solution of No. 5 of the 
Andes rocks. This contained 9°17 x 10—” gram radium. To 
this was added 7:07x10-” gram radium from standard 
solution. Closure 21 days. 


Total radium = 16°24 x 10-” gram. 
Gain = 19 scale-divisions per hour. 


Hence ge Lo: 2441052 — 85 y 10-2. 
19 

The use of the slow admission capillaries used in the 
foregoing experiments was continued throughout the rock 
tests, although comparative experiments in which the ad- 
mission was made as rapid as was consistent with the safety 
of the gold leaf, and again with the capillary tube, showed 
no detectable difference. The construction of the electro- 
scopes were of course also preserved in all particulars alike. 

The importance of conducting the calibrating experiment 
under conditions of the solution as nearly identical as 
possible with those under which the actual experiments are 
made, seems immediately apparent from these results. Such 
a variation in the constant as appears in the above experi- 
ments, in the case of electroscope A, might introduce an 
error of deficiency of as much as 3() per cent. into each 
experiment. 


Effect of Sunlight. 


Care was taken to shield the electroscopes from bright 
daylight, and the discharge rates were read throughout in 
semi-darkness. A short series of observations showed that 
the discharge rate might be raised from 5 scale-divisions 
per hour—the normal leak—to over 80 per hour in direct 
sunlight, and 30 per hour in bright diffused light. The 
normal leaks of 10 and 5 scale-divisions per hour for “ A ” 
and “ B” respectively, were wonderfully constant from day 
to day, scarcely ever varying over 1 scale-division per hour, 
three hours after refillmg with fresh air. Under these 
circumstances an increase of 2 or 3 scale-divisions per hour, 
at an interval of three hours subsequent to the introduction 
of the emanation, was unmistakably evident. 

This would be accounted for by as small a quantity as 
3x°8x10-” gram of radium in a solution containing say 
10 grams of rock. Hence a quantity of radium of about 
"24x 10-1? gram per gram was capable of measurement 
with a fair degree of accuracy. 

The method of extraction of the emanation was essentially 
that described by Professor the Hon. R. J, Strutt, with the 


42 Mr. A. L. Fletcher on the Radioactivity of 


modification adopted by Professor Joly, of boiling—tfor 
thirty minutes—with tale, in vacuo. A slight departure 
from the method described by Professor Joly was made. 
At the conclusion of the boiling, the cooling water was cut 
off, and the ebullition accelerated until steam began to 
condense in the receiver bulb, when the gas was cut off, and 
the pinch-cock closed simultaneously. In addition to the 
possible advantage of a brief violent ebullition, this had the 
effect of removing any emanation from the flask and con- 
denser, without the addition of any water to the solution ; 
which was then ready for the estimation of contained thorium. 
A slow transference of the emanation to the electroscopes 
was then effected through the capillaries—the process 
occupying about 10 minutes. A glass tube, containing a 
water-bubble, served to indicate the moment when the electro- 
scope was filled to atmospheric pressure. 


The Rocks. 


The following representative specimens of the rocks 

. =) . ° * 

dealt with, were selected, and submitted to microscopie 
examination :— 


Specimen 1.—Large phenocrysts of both soda-lime and 
lime-soda felspars with rare sanidine, in a turbid ferruginous 
ground-mass with small tabular felspars, and brownish 
microliths. Magnetite and red oxide of iron abundant. 
Many indeterminate iron-stained crystals. No glass. Much 
augite in yellowish-green phenocrysts. Borders often ferru- 
ginous. (General appearance of rock highly altered. External 
colour light grey. Structure trachytic. 

An altered Trachyte, with some Andesitic characteristics. 


Specimen 4.—Ground-mass consisting of grains of felspar, 
and turbid glass with indeterminate ferruginous particles. 
The whole stained red-brown with iron oxide. Many frag- 
ments of andesitic lava. External colour brown. 


A Felspathic Tuff, much altered. 


Specumen 7.—A fine-grained homogeneous ground-mass, 
consisting mostly of tabular felspar—sometimes banded— 
with some turbid glass. Considerable calcitic matter de- 
veloped. Jron-stained particles common. External colour 
red. 

A fine-grained, altered, Andesitic Tuff. 


_ Specimen 8.—Many twinned phenocrysts of. soda-lime and 


the Rocks of the Transandine Tunnel. 43 


lime-soda felspars—oligoclase predominant—in a ground- 
mass of oligoclase microliths. Magnetite in small grains 
dusted over the field. No glass, but areas stained green and 
eryptocrystalline. Colour grey. 

An Oligoclase-Trachyte, somewhat decomposed. 


Specimen 12.—Phenocrysts of tabular and columnar potash 
and soda lime felspars, in a ground-mass in which small 
oligoclase felspars predominate over crystals of soda-lime 
felspar. Some augite crystals. No glass. Much iron oxide 
in grains, and irregular areas. Colour red-brown. 

An altered Trachyvte. 


Specimen 13.—A coarse-grained, dark brown rock, partially 
disintegrated. Consists of irregular grains of different lavas 
cemented in a nearly opaque matrix, much clouded by iron 
oxide. A few grains of olivine. 


An altered, basic Tuff. 


Specimen 14.—A few phenocrysts of columnar lime-soda 
felspars, and sanidine, in a ground-mass consisting of oligo- 
clase microliths with much calcite in large areas and irregular 
cracks. No glass; but much green chloritic matter de- 
veloped—possibly altered glass. Some grains of magnetite. 
Colour medium grey. Grain fine. 

A partially decomposed Sanidine-Oligoclase-Trachyte. 


Specimen 17.—Consists of andesitic fragments with calcitic 
and ferruginous alteration products in a felspathic ground- 
mass. All constituents in various stages of decomposition. 
Colour light. Grain medium. 

A decomposed Felspathic Tuff. 


Specimen 18.—Consists of columnar phenocrysts of sanidine 
and oligoclase, with more basic felspars, in a ground-mass of 
oligoclase microliths and calcareous matter. Much chloritic 
and calcitic matter developed throughout. No magnetite. 
No glass. Colour light grey. 

A Sanidine-Oligoclase-Trachyte, partly decomposed. 


It was not considered necessary to examine microscopically 
the remainder of the specimens, which have been named by 
comparison with those selected. 

The following table shows the radioactivity of the speci- 
mens determined. The thorium content was estimated from 
the same solutions by Professor Joly, using the method 


described by him (Phil. Mag. May & J uly, L900): 


44, Radioactivity of Rocks of the Transandine Tunnel. 


£2 | 'o8 jie 
‘ aes Ren x 0 
feats ie x aN gs State of 
No. Description. 8 e g 5 E 3 Solute 
si | 38 | Se 
ns PG Sb Fi & 
ERS re aU 2°425 32 39 
Pe MOVE 2°425 80 43 | 
Go| MOIEORERECEOC Soe. cece sane. 2°500 40 0 
AV Pelspaime Pade ol... 0. ke. | 2°650 1:14 05 
Dee MM EERIE! Comes seid ww eie wes 8 | 2-630) 1°40 -— | 
SSL ee | 2420 1:26 “80 
Tie) inc ei Galli rr | 2°850 “90 — 
8. | Oligoclase Trachyte ...............| . 2°890 35 ‘58 Dp. 
ENRAGED LOS) 2. Lo. lee.. es Sted wee es 3°125 WT “OL 
PUPP EUEREAEC i. iin. s.ies- 0) see. dgaeceen | 3158 OT ‘Ol D. 
MMB NIIIGSILC 06.00: ca cen canine vps arses 3°200 "52 “41 | 
Be ermal 02 kee. .c..ccoaeteley 3360 | “64 a7 | 
SS ee oie ois ain baseman ance enne | 3°466 Td "84 p | 
14. | Sanidine-Oligoclase-Trachyte.... 3°540 1-07 “Til p | 
Mea M TMS EDEL rat Said ds vals's's a n¢seneowmeeert | 3°545 68 "85 p | 
MMU msc <5 +<= poip Hawsus Hees seeauaane 4 209 "93 111 Dp | 
ee eelsmabhic Taft 2... an <ssimweeseiees 4-409 1:38 DD p | 
18. | Sanidine-Oligoclase-Trachyte ..., 3°609 "33 ‘30 . 
EP IHAL ss #2 ava caey's ch-owewaeee dae sam | 4709 "58 "8S p | 
| MEAN o....... "79 | 56 


It may be noted that the mean ratio borne by the thorium 
content of these rocks to the radium content, viz. *71 x 10%, 
bears an approximation to that obtained by Professor Joly 
in a paper on “The Radioactivity of certain Lavas” (Phil. 
Mag. October, 1909), where the proportion of thorium to 
radium found was °65 x 10’. 

The striking feature of these results is their poorness in 
radioactive matter. This may possibly be referred to the 
alterations undergone by the rocks, which may have been 
attended by the removal of all that part of the radium 
which was soluble in percolating waters. In no case was 
any trace of radium, and in few cases was any thorium 
discovered in the alkaline solutions. 

The quantities of radioactive matter observed would give 
rise to very small heating effects. Taking the mean radium 
content for the recks in the neighbourhood of the tunnel as 
‘79x 10- gram per gram, and the thorium content as 
-52x10-° gram per gram, we find for the radium the rate 
of evolution of heat as 201°6 x °79 x 10-”=1°6 x J0-”° calorie 
per gram per hour ; and for the thorium—allowing that the 
heat produced per hour per gram of elemental thorium in 


Reflected-Difiracted and Diffracted-Refected Rays. 45 


equilibrium as 2°38 x 10-° calorie per bour—we get for the 
heat production 2°38 x 10-° x 56 x 10-°=1°33 x 10-" calorie 
per gram per hour. The total heat evolved is therefore 
2°9 x 10-1° calorie per gram per hour. 

The small thermal effect due to such low quantities of 
radioactive matter in rocks, has been pointed out by Professor 
C. H. Lees (Roy. Soc. Proc. A. vol. Ixxxiil. p. 344), and we 
should not expect—nor was there found—any abnormally 
high degree of temperature in the tunnel. 

In conclusion I desire to express my gratitude to Professor 
Joly, at whose suggestion and under whose directions the 
work was carried out, and to Professor W. E. Thrift, for his 
kindness in providing the use of the room in which the 
experiments were performed. 

Geological Laboratory, 


Trinity Colleye, Dublin. 
May, 1910. 


Ill. The Interference of the Reflected-Diffracted and the 
Diffracted-Reflected Rays of a Plane Transparent Grating. 
and on an Interferometer. ByC. Barus and M. Barus”. 


Wee a ee 
1. Introductory. ‘dae | i 


io parallel light, falling on the front face of a transparent 
plane grating, is observed through a telescope after 
reflexion from a rear parallel face (see fig. 1), the spectrum 
is frequently found to be intersected by strong vertical inter- 
ference bands. Almost any type of grating will suffice, 
including the admirable replicas now available, like those of 
Mr. Ives. In the latter case one would be inclined to refer 
the phenomenon to the film and give it no further con- 
sideration. On closer inspection, however, it appears that 
the strongest fringes certainly have a different origin and 
depend essentially on the reflecting face behind the grating. 
Jf for instance this face is blurred by attaching a piece of 
rough wet paper, or by pasting the face of a prism upon it 
with water, so as to remove most of the reflected light, the 
fringes all but disappear. Ifa metal mirror is forced against 
the rear glass face, whereby a half wave-length is lost at the 
mirror but not at the glass face in contact, the fringes are 
impaired, making a rather interesting experiment. With 
homogeneous light the fringes of the film itself appear to the 
naked eye as they are usually very large by comparison. 


* Communicated by the Authors. Abbreviated from a report to the 
Carnegie Institute of Washington, D.C., U.S.A. 


46 Messrs. C. and M. Barus on the Interference of 


Granting that the fringes in question depend upon the 
reflecting surface behind the grating, they must move if the 
distance between them is varied. Consequently a phenomenon 
so easily produced and controlled is of much greater interest 
in relation to micrometric measurements than at first appears, 
and we have for this reason given it detailed treatment. It 
has the great advantage of not needing monochromatic light 
and of being applicable for any waye- “length whatever, and 
admitting of the measurement of small horizontal angles. 

When the phenomenon as a whole is carefully studied it is 
found to be multiple in character. In each order of spectrum 
there are ditterent groups of fringes of different angular sizes 
and usually in very different focal planes. Some of these are 
associated with parallel light, others with divergent or con- 
vergent light, so that a telescope is essential to bring out the 
successive groups in their entirety. At any deviation the 
diffracted light is necessarily monochromatic ; but the fringes 
need not and rarely do appear in focus with the solar spectrum. 
If the slit of the spectroscope is purposely slightly inclined 
to the lines of the grating certain of the fringes may appear 
inclined in one way y and others in the opposite way, producing 
a cross pattern like a pantograph. The reason for this 
appears in the equations. 

In any case the final evidence is given when the reflecting 
face behind the grating is movable parallel to it. The inter- 
ferometer so obtained is subject to the equation (air space e, 
wave-length A, angle of incidence 2, of diffraction 0’), 

d¢=A/2(cos 0’ — cos), 
and is therefore less unique as an absolute instrument than 
Michelson’s classic apparatus or the device of Fabry and 
Perot. Its sensitiveness per fringe, de, depends essentially 
upon the angle of incidence and diffraction and it admits of 
but 1 cm. (about) of air space between grating face and 
mirror before the fringes become too fine to be available. 
But on the other hand it does not require monochromatic 
light (a Welsbach burner suffices), it does not require optical 
plate glass, it is sufficient to use but a square cm. of grating 
film, and it admits of very easy manipulation, for painstaking 
adjustments as to normality, &c., are superfluous. In fact, it 
is only necessary to put the sodium lines in the spectrum 
reflected from the grating and from the mirror into coin- 
cidence both horizontally and vertically with the usual three 
adjustment screws on grating and mirror. Naturally sun- 
light is here desirable. Thereupon the fringes will usually 
appear and may be sharply adjusted upon a second trial 
at once. 


Reflected-Difiracted and Diffracted-Reflected Rays. 47 


When the air space is small, coarse and fine fringes (fluted 
fringes) are simultaneously in focus, one of which may be 
used as a coarse adjustment on the other. Tinally, the 
sensitiveness per fringe to be obtained is easily a length of 
one half wave-length in the fine fringes and one wave-length 
in the coarse fringes, though the latter may also be increased 
almost to the limit of the former. 


2. Observations. 


The following observations were made merely to corrohorate 
the equations used. The general character of the results 
will become clear on consulting the following abbreviated 
table chosen at random from many similar data. An Ives 
replica grating with 15,000 lines to the inch (film between 
plates of glass -46 cm. thick) was mounted as usual on a 
spectrometer admitting of an angular measurement within 
one minute of arc. Parallel light fell on the grating, fig. 1, 
gg, under different angles of incidence, 2, and the spectrum 
lines were observed by reflexion (after reflexion from gg and 
the rear face /f) at an angle of diffraction @ in air, both in 
the first and second order of spectra, and so far as possible 
on both sides of the directly reflected beam. In view of the 
front plate, the angle i corresponds to an angle of refraction 
7 within the glass, and the angle @’ similarly to an angle of 
diffraction 0, respectively. Hence r>0, or @,<,r denotes the 
sides of the ordinary ray on which observation is made. As 
a rule these were as nearly as possible in the region of the 
D line passing toward H. Finally, 60 denotes the angle 
between two consecutive dark fringes, observed and computed 
as specified. Similarly de will be reserved for changes of 
thickness e of the glass and de’ for changes of the air space 
in case of an auxiliary mirror MM, 

For 1=0° the number of groups of lines was a single set 
in each order, but only the end of the spectrum could be 
seen. Measurements refer (about) tothe Cline. For i=45° 
several groups were too close together, or too faint for 
measurement, and the same is true for 2=22°5. An estimate 
of divergence is all that could be attempted on the given 
spectrometer. ‘The case 0,>~r was usually not available, but 
for 1=22°°5 two sets were found in the first order, one being 
the normal set. ‘The fringes in all cases decrease in size from 
red to violet, but less rapidly than wave-length (§ 7). 

Whether they are convergent or divergent for a given set 
of fringes, as for instance for the strong set, depends on the 
position of the grating. Thus the divergent rays become 
convergent when the grating is rotated 180° about its normal. 


48 Messrs. C. and M. Barus on the Interference of 


It is therefore definitely wedge-shaped. In fact when the 
auxiliary mirror JZ is used, the fringes may be put anywhere, 
either in front of or behind the principal focal plane, by 
suitably inclining the mirror. 
3. Equations. 
If we suppose the film of the grating gg to be sandwiched 
in between plates of glass each of thickness e, it will be seen 


Kia) 


M 


ve. 
ET e 


e 


that triplicate rays pass in the direction t, (0,;/>7), or of 
ty (@,'<2), which will necessarily produce interference either 
partial or total. With respect to ¢,, the only light received 
comes either from LD, by direct diffraction at gg, or from 
RD, by reflexion from the lower face ff, and thereafter by 
diffraction at gg; or from Di, by diffraction at gg and 
reflexion at ff. Similarly, the light along t, comes in like 
manner either from 2), or DR, or RD, With regard to 
the angles of incidence and refraction or of diffraction within 
the glass or outside of it, we have the equations for the first 
and second order of spectra (D being the grating space). 


Sy = psn, te ee , 
si G,' = e810 64,6 kw eee +. 
Bim 6 LSI Oy. Jue. ae er 
sine — sinf,=A/Dy or = 2)/Du, ees 
sin 6;— sin r =A/Dy or =2A/Dp, ae 
sine — sin 0, =A/D, ete, 472.) 2. rr 


sin @,/— sini =2/D,, etc) % . se 


49 


Rejflected-Diffracted and Diffracted-Reflected Rays. 


TaBLe I.—InTERFERENCES. Grating between plates of glass each ‘48 em. thick. Additional rear 


Observed Computed 
Side. Order, | Colour. Rays. hi Mean 6’, 60’ 60! Remarks, 
minutes, minutes. 
eg 6 0 iL #0, Convergent. 0° 24° 8! 137 Let 
Jes Wide: 4S i 2 eae omen « Chie Couatg rae cara About double di- 
vergence but not 
| clear. - 
r>0. Sale D to H.| Less convergent. 45° 21° 1! 1’°80 170 Strong, Total > © 
m2? 8b. 1 Dto E.| Parallel. 45° PBN e 3 re Hstimated **, 
O=18° 35% 1 Do. Divergent. 45° 21° 32' uF | Atta Hstimated **. 
1 Do, More divergent. 45° 22° 40! "87 
r>0. y- Wo Convergent. 45° 31’ 31°32 ° seitpanaae Clear but faint, 
2 Do, Convergent. 45° samme 8 sven anteR Very fine and close. 
2 Do. Parallel. 45° 31! 1:20 
2 Do. Divergent. 45° saat sates aah Not clear, 66’=2'-0? 
2 Do. Very divergent. 45° Pe ee CA aus Not clear, 60'=3"0? 
pm, 1 | DtoE.| Very divergent. 22°°5 1° 380’ eae ieMbdetas Faint but certain. 
ee as 1 Do. Divergent. 22°°5 cians sigan sama Faint, 60/=2'? 
eS a 1 Do. Nearly parallel. 22°°5 Rade af agtaie ei ttapgnat Faint, 60'=1''5? 
1 Do. Parallel. 22°°5 «stig ree ate Fine, close, and strong. 
3 ] Do. Convergent, BOSS 2/4 72°°0 1°85 ce: Strong. Total. 
Interferometer. Thicknesses e and e’ of glass. 
i D to E. Diceuccte 22°°5 1'-83 1'-84 e='48cem. Total. 
1 | Do. Fashigne Ne 22°) | 15 | 115 e= "(7 om. Lotalk 


plate (mirror) *29 em. thick. 


* Colour not definite, only end of red spectrum seen. 


Index of refraction of glass w==1°527. 


** Lines strong but too fine and close together, 


50 Messrs. C. and M. Barus on Interference of 


where w is the index of refraction of the glass, found to be 
equal to 1°5265 for sodium light, by breaking off a small 
corner of the glass of the grating and using Kohlrausch’s 
total reflectometer. 

If the wave fronts be taken in the glass plate ffgg, the 
equations become 


WN = Zep Cosy, “or. >). 
mK = 2ewcos@,, «s.r 
nr = 2eu(cosr—cos 0)». « ier 


with three other corresponding forms for 0< 7. 
For an air space between gg and M/ the equations wou'd be 


nr = 2e cost, 
nr = 2ecos 6’, 


nr = 2e(cos 6'— cos2); ke. 


4, Differential Equations. 


The quantity measured on the spectrometer is essentially 
angular and preferably d@'/dn, the angular distance apart of 
the fringes, in radians. Later we shall measure 6e or the 
linear displacement of the parallel faces per fringe. In any 
measurement, however, we meet with embarrassment, inas- 
much as n, A, w, 7”, O, 6’, are all variable. The angle z and 
the thickness e and the grating space J) are alone given. 
Among these the variation of 7 with w and A must be found 
by experiment. JT ortunately, in case of the interferometer, 
all these variables are eliminated and e alone changes subject 
to a given i and @’. The mw used need not be known. 
See § 7. 

For the present purpose, as the variation of mu enters only 
as a correction, we have been satisfied with the usual results 
in physical tables. If from the C to the D line 


(dys|) /(An/r) = —"016, 
and from the B to the C line, = —‘013, we may write 


and therefore 


We shall abbreviate a=-015, b=1 +a. 


ae pe 


> 


Reflected-Digfracted and Diffracted-Reflected Rays. 951 


The case 9>r in the present paper is not of much experi- 
mental interest. We may therefore omit it here. For the 
case of r>6 we shall have successively, and for the total 
interferences RD, DR, equation (8), 


dX! Xeosd)) dO (11) 
ae sinr—6sin 6 dn’ ; 
_a& _, tanrcos? dé AS keane 


dn sinr—bsin @ dn 


du _ _apcos 0 dé 
dn sinr—bsin®@ dn’ 


d@’ _ « cos O(sin r—sin @) dé 
dn cos (sin r—b sin 8) dn’ 


(13) 


(14) 


and finally corresponding to equations (6), (7), (8), 


dé’ cos @-_-XA(sinr — sin 8) 
dn 2ecos@’ b—sinrsin®g ” 


(15) 


dd’... cos8 (sin r—sin 0) 16 
dn 2ecos@' beosrcos 6+asinr tan r cos 8’ (16) 
d's cos 8 d(sin 7 —sin 8) 


dn ~ Be cos 8 b(1—cos(—8)) +asinrsinO(—cotBtaary 7) 


the last term in the denominator being corrective. Here 
d’/dn is the observed angular deviation of two consecutive 
fringes. 


5. Normal Incidence or Diffraction, &e. 


For the case of normal incidence :=r=0, the equations 
corresponding to (6), (7), and (8) take a simplified form, and 
are respectively 


ay COs t ; =f 
emia oe 
age Zt 1 ks f 
aor ne) 


_ dO, _ _cos@ Asing , ; 
dn ~ 2be cos 6’ 1—cos 6 pie ak eee 


If 6’=0=0 for normal diffraction, which is particularly 
EK 2 


52 Messrs C. and M. Barus on Interference of 


useful in Rowland’s adjustments as well as on the spectro- 
meter 


vb aie geld sin 7 
Fal e—o Ze b(1—cos 7) —a sine tanr 


for the case of total interference corresponding to equations 


(8) and (17). Iti=—@' or r=—8, 


.. edo”, Sieh AES 
at 2e tan 8 cos 0" 


6. Comparison of the Equations of Total Interference with 
Observation. 


The partial interferences corresponding to equations (6) and 
(7) are usually too fine to be seen unless ¢ is very small. They 
amount in cases of equations (15) and (16), where e="48 cm., 
to the following small angles :— 


(15) (16) 
i=0°, d0'/dn=0"-060, | dé@'/dn=O™0ne: 


29°5, 0/048, 0! 050; 
45°, 0/-057, 0-058; 


usually less than four seconds of are and are therefore lost. 
The origin of the fine interferences actually seen in the table 
is thus still open to surmise. With small e and the inter- 
ferometer they are obvious. 

The total interferences as computed in the above table 
agree with the observations to much within ‘1 minute of arc, 
and these are experimental errors ; particularly so, as it was 
not possible to use both verniers of the spectrometer. The 
interesting feature of the experiment and calculation is this, 
that 60’ has about the same value for all incidences i from 0° 
to 45° and even beyond. The equations do not show this at 
once owing to the entrance of w and r. But apart from a 
and b, equation 17 is nearly 


dO L.A aa: 
dn ew 1—cos (r—8)’ 


which is independent of 7 to the extent in which cos (r— 6) 
is constant. The dependence of d6’/dn on wave-length is 
borne out. See § 7. 

Finally, d0’/dn is independent of yw, except as it occurs in 
a and 0. | 


Reflected-Difracted and Diffracted-Reflected Rays. 53 


If the glass plate fgg is removed and a mirror JM used, 
as in the interferometer, the fringes may be enormously 
enlarged by decreasing e and the measurements made with 
any degree of accuracy; but such measurements were 
originally impracticable and have now little further interest. 


7. Interferometer. 


The final test of the above equation is given by the last 
part of the table for different thicknesses of glass, e=°48 
and e="77 cm. The results are in perfect accord. 

These data suffice to state the outlook for the interferometer. 
In this case n and e are the only variables, so that equation (8) 
becomes 


6e=)/2u(cos @—cos 7), 


where de is the thickness of glass corresponding to the 
passage of one fringe across the cross-hairs of the telescope. 

If instead of glass in the grating above, an air space inter- 
venes between the film of the grating and the auxiliary 
mirror M (fig. 1), the equation reduces to 


r 
EE Ce Ma ew) 
where i and 6’ are the angles of incidence and diffraction in 
air. 

These equations (20) embody a curious circumstance. 
Inasmuch as @ and @ change asi increases from 0° to 90° 
from negative to positive values at about 2=13° and 7=20°, 
respectively, the denominator of either equation (20) will 
pass through infinity (for air at about z=10°). Hence at this 
value of 2 the motion of the mirror W produces no e-effect 
(stationary fringes), while on either side of it the fringes 
travel in opposite directions in the telescope when e changes 
by the same amount. In the negative case the sensitiveness 
for air spaces passes from de=—-*000,489 to de=—m per 
fringe. In the positive case from ée=+a to de—:000,039 
per fringe, or to a limit of about a half wave-length in case 
of 15,000 lines to the inch. This limiting sensitiveness may 
be regarded as practically reached even at 1=40°, where 
de="000,155 cm. per fringe and an angle of about 1=45° is 
most convenient in practice. 

The addition to the large fringes the fine set appears 
when e is small or not more than a few tenths of a milli- 
metre. The sensitiveness of these is naturally much more 


ice. > eee i = eR id <r _ — 


Soo SS SR a 


Sea a Se a ee NE ee a ee ee Se a : — = 


: = ee k= 


o4 Messrs. C. and M. Barus on Interference of 
marked. In the two cases 


66 = A/2 cost. 5 ©» = 16 e 
and 


Se = 0/2cos 0’, . : «ms a 


so that nearly X/2 per fringe is easily attained. 

At i=20° about, and in case of an air space, 6’ is nearly 0°. 
We suggested above that these fine fringes may be used as a 
fine adjustment in connexion with the large fringes, on 
which they are superimposed. In appearance these large 
fluted fringes are exceedingly beautiful. The fine fringes 
have the limiting sensitiveness of the coarse fringes, 2. e. the 
cases for2=90° or @’ equal to maximum value. If in different 
focal planes, both sets of fine fringes may be seen separately 
for small e (air wedge). 

Hquation (20) shows that for smaller grating spaces, D, and 
consequently also in the second order of spectra, there must 
be greater sensitiveness, cat. par.; but as a rule we have 
not found these fringes as sharp and useful as those in the 
first order. 

The limiting sensitiveness per fringe, however, follows a 
very curious rule. If in equation (20) we put i= 90°, 


28e=Xr/ V/r(2Q—r) 
in the first order ; if r=A/D, and 
25e=r/2 V7r(1 —7) 


in the second order. J) is the grating space. Both equations 
have a minimum, 6e=2/2, at A/D=1 in the first order and 
r/D="'5 in the second order, beyond which it would be 
disadvantageous to decrease the grating space. These mini- 
mum conditions are as good as reached even when JD corre- 
sponds to 15,000 lines to the inch, as above, where roughly 

10%Se=88 cm. in the first order and 10%e=33 cm. in the 
second order. 

To view the stationary fringes of the first order was on 
practicable since they occurred for 2=10°, whereas the tele- 
scopes were in contact at about 20°. In the second order of 
spectra they may be approached more nearly, as they occur 
when 2 is roughly 20°. If the distance e is made small 
enough so that the three cases of equations (20), (20’), (20) 
are visible, the appearance is very peculiar. The fringes of 
equation (20) are very slow moving. They are intersected by 
the small fringes of equation (20'), producing the fluted pattern 
already discussed. Over all travel the rapidly moving fringes 


Reflected- Diffracted and Diffracted-Reflected Rays. 55 


of equation (20"), producing a kind of alternation or flickering 
which it is very difficult to analyse or interpret until e is 
very small, when all three sets are broad and easily recognized. 
Sunlight should be used. Nothing like these alternating 
fringes is seen in the first order. 

The above equation shows finally that de is not exactly 
proportional to wave-length, though the former decreases 
with the latter as found above. 

The three equations (20) indicate finally that for 1 >0' 
all fringes travel in the same direction with increasing e ; 
whereas if 0’ >i, the set corresponding to equation (20) travel 
in a direction opposite to that of the set (20') and (20”). This 
is strikingly borne out by making the experiment for 6! >: 
with a small angle 7, both in the first and second order. 
Flickering in the latter case is accentuated. 

Table iL. contains a few data obtained by carrying the 
mirror on a Fraunhofer micrometer, reading to ‘0001 cm., 
toward a stationary grating film. Observations were made 
in the region of the D lines. The grating was originally 
between plates of glass e="48 em. thick. Finally the plate 
between grating and mirror was removed, the whole distance 
now being an air space. This has no effect on de, but e may 
then be reduced to zero and the fringes enlarged. 


Tas.E [].—Interferometer Measurements. Replica grating 
(collodion film). Air space 0-25 cm.; total space 
O—"9 cm. 


| | Coarse fringes. Fine fringes. 
a. | 6’. | | 
| gex0® | dex10® | dex10® | sex10% 
obs. | gale. obs. | cale. 
a SS ai 3k HD ae, eee ane 
Phe exes Se ea cm. em. em. | em. 
ip james) or 390 391 == . — 
| 
Ab» iO ANA? 7 131 130 | — — 
| 
67 30 So ek 72 68 | — = 
By SO va a | eae Wis SRNL) Lp ee | im 
a0) 2 S37 241 240 | — — 


Glass removed between mirror and grating. 


. aed Ree crate of 34 ies 
45 0 |.20 9 129 12 |.) age aii 


56 Messrs. C. and M. Barus on Interference of 


These data merely test the equations, as no special pains 
were taken for accurate measurement, which neither the 
micrometer screw nor the special adjustments warranted. 
Usually the micrometer equivalent of 50 fringes was 
observed on the screw. The maximum distance e between 
grating and mirror was ‘48 cm. of glass and ‘25 em. of air 
conjointly, or within 1 ecm. In the case of fine fringes mere 
pressure on table or screw impaired the adjustment. 


8-9. Secondary Interferences. 

We come now to the consideration of the minor inter- 
ferences (Table J.), which are either weaker, finer, or more 
diffuse than those discussed. In the interpretation of these 
we have not met with adequate suecess (assuming that after 
two reflexions the fringes can no longer be seen) to give it 
space here. We will therefore dismiss it with the remark 
that each of the three incident rays of figure 1 corresponds 
to three emergent rays for 6>7r and three for 0<7r. If we 
call these a, 6, c, a’, b', c', a”, b", ce” for either case, the possible 
partial interferences may be found by grouping the terms of 
the following determinant in pairs :— 


7 / 
| a’ 6” e / 


There are 18 cases, most of which, however, are identical 
in path-difference. 


10. Convergent and Divergent Rays. 


What finally characterizes the above groups of inter- 
ferences is the difference in position of their tocal planes. 
They rarely coincide with the spectrum (parallel rays), and 
hence do not always destroy it. If present with the spectrum 
the latter is wholly wiped out. If the strong fringes are 
convergent for a given adjustment of grating they become 
divergent when the grating is rotated 180° about its normal. 
Hence the plates of glass are sharply wedge-shaped, and to 
these differences the location of the focal planes is to be 
referred. 

In addition to this the three regular reflexions are not in 
the same focus which shows the surfaces (collodion film) to 
be slightly curved. The above experiments succeed best 
when two of the reflexions are yellowish, which probably 
means that the grating face is from the observer. 


Reflected-Digrracted and Dif'racted-Reflected Rays. 57 


Suppose the remote glass face makes an angle dr/2 with 
the surface of the grating. Then the DR ray of the strong 
interferences has its angle increased by dO=dr, whereas the 
AD ray receives an increment of but 


cos? 
— > Sa adr. 
cos 0 


Hence if the DR and RD rays were parallel for parallel 
surfaces, they would be at an angle corresponding to 


dé 


apiars cos @ ; 


where dr/2 is the angle of the wedge. If DA is negative in 
character, opposite conditions will hold, since dr and d@ 
change signs together. 

Rays all but paraliel will cross each other in front 
(convergent) or behind (divergent) the grating, depending 
on their mutual lateral positions. As a ray moves from the 
right to the left of the normal, the phenomenon may change 
from divergence to convergence, and vice versa. 

These relations are very well brought out by the inter- 
ferometer of which the mirror MM may be inclined at pleasure. 
If small values of deviation only are in question, this instru- 
ment becomes a means of measuring small horizontal angles 
y between mirror and grating as these involve less change of 

ocus. 

In fact, if A is the vertical height of the illumination at 
the mirror 1, and the corresponding obliquity of fringes is 
equivalent to an excess of V fringes crossing the bottom of 
the cross-hairs as compared with the top for a wave-length i, 


y= Noe/h ; or 


d@—dr ecosr—cos8 Hi, ben hte 


VX 


~~ 2h(cos 6’ —cos ai 


q 


The question next at issue is thus the value of h. It will 
be noticed that if parallel rays fall upon the slit, they will be 
brought to a focus by the collimator objective first, and 
thereafter by the telescope objective placed at a diametral 
distance D beyond it. Then if S is the-vertical length of 
slit used, and 7, and /; the focal lengths of the two objectives 
respectively, it follows that the length h=S is virtually 
illuminated. Hence 

Nn 
3 F285 (cosi8. Cos) 0, ee. 


For since the angle y, or a ratio, is in question, Née/h is 


08 = Reflected-Diffracted and Dif'racted-Reflected Rays. 


constant, and it makes no difference where the mirror M@ may 
be placed, 2. e. how great the absolute vertical height of the 
illumination h may be. 

In case of this method (parallel light impinging on the 
slit) the illumination at each point of the image is received 
from but a single point (nearly) of the mirror, whereas if the 
light falling on the slit is convergent, the whole vertical 
extent of the mirror illumination contributes to each point of 
the image in the ocular. Hence in the latter case the fringes 
are only sharp when JZ and the grating are rigorously 
parallel, and they soon become blurred when this is in- 
creasingly less true. The same observation also accounts 
for the greater difficulty in adjustment when lamp-light is 
used. In any case, equation (25) furnishes V/S. NV may be 
obtained with an ocular micrometer. The angle y may also 
be found by actually measuring the inclination to the vertical, 
8, ot the fringes in the ocular. Here if the height of image 
s in the ocular corresponds to the vertical length of slit 8, 


z =f 2(1-=); 26 
SS) ie on ‘Pla i May 
while 
_ Wide’ 
seed ae" 


where d6'/dn is given by equation (17). Hence s may be 
eliminated and 
iy: 
30 
fe 8 tag 
If, now, we further eliminate V/S in equation (25) by 
equation (27), we have finally 


ae BM get 20D Se) 
Me 2f. fi(cos 0’ —cos 1) d0'/dn’ 


so that y is given in terms of 8, the observed inclination of 
fringes in the ocular. To measure 8 the ocular must be 
revolvable on its axis so that the cross-hairs may be brought 
into coincidence with the fringes, and the angles found. To 
measure JV, the D lines, if in focus, may often be used for 
reference in place of vertical cross-hairs, as they remain 
vertical. 


Lengths of the Focal Lines of Cylindrical Lenses. 59 
Using the data of the above experiments, if ~=45°, V=1, 
fe=fr=D (nearly)=23 cm:,' cos @’—cos2= "2264, 


S= "9 em. rA=60 x LO~*, 
ag jdn=4A93 x 10-8, 
whence 


y= 1G 10; radian, 


or about a half minute of are per fringe, and 6=44' per 
fringe. Thus 8 is about 88 times as largeas y. At i= 22°°5, 
y=1':5 per fringe, @B=45' per fringe. Naturally the sensi- 
tiveness increases with the angle ot incidence. When the 
fringes are large 1/10 fringe is easily estimated, so that a 
horizontal angle y of a few seconds between mirror and 
grating should be measurable. An ocular micrometer as 
suggested would carry the precision beyond this. 


Brown University, 
Providence, R.I. artis 


F ; ie, “ay F “ 
pon She May Wig : 5 
j . 2 NDR CONE en a A 
2 LSE eee REA Ee Serene PRI PN? aap nea abo) See, Ly ess es Lee aes SCONE EY, 
$ omesngh my fevarens 4 i arian, e % 


IV. On the Lengths of the Focal Lines of Cylindrical Lenses. 
By A. Wuitwew., M.A., A.R.C.Se1.* 


NHE following paper is a continuation of one entitled 
“On Refraction at a Cylindrical Surface,” published 
in the Phil. Mag. for July 1903 ¢. In that paper the form of 
the focal lines or focal areas produced by refraction at a 
cylindrical surface was investigated, the aperture parallel to 
the axis of the cylinder being considered to be unlimited. 
The object of the present paper is to find the lengths of 
the focal lines produced by a single cylindrical surface or by 
one or more cylindrical or sphero-cylindrical lenses, the 
aperture being so small that the focal lines may be con- 
sidered to be straight lines. The formule arrived at are 
analogous to the ordinary first-approximation formule. for 
thin spherical lenses. - 


* Communicated by the Author. 
+ There is an error in this paper on p. 54. The two equations at the 
foot of this page should be 


Os CAAT Ee 
d+Va?+h2 a 


a" = Via—rE ED — (4-7), 


60 Mr. A. Whitwell on the Lengths of the 
1. Yo find the lengths of the focal lines of a single 


cylindrical surface. 
Fig. 1 represents an elevation and fig. 2 a plan of the 


system of rays. The light is supposed to pass from left to 
right and the index of refraction of the second medium = yp. 


Distances measured to the right of 0 or o are positive. The 
axis of the cylindrical surface is vertical and the are b, 0, a, 
represents the trace of the surface on the horizontal plane. 
h, = ola! = the semi-aperture parallel to the axis of the 
cylinder or the axial semi-aperture. 
h, = oa = ob = the semi-aperture at right angles to the 
axis of the cylinder or the tangential semi-aperture. 
ac’ is the elevation and ac, be, the plans of two normals 
to the cylindrical surface. 
1, = the radius of the surface. 
u = o’d’ = the distance from o’ of the point in which the 
incident rays cut the axis of z. 


(a) The line parallel to the axis of the cylinder; this may 
be called the axial focal line. 


Two incident rays symmetrical with respect to the plano 
of fig. 1 are represented in elevation by the lines a’d’ and 
in plan by the lines ad, bd; the corresponding refracted rays 
are represented in elevation by the line a’ fg’, and in plan 
by the lines aeh and bek. 

The incident ray a’d’, the refracted ray a’o’, and the 
normal a'c’ are all in one plane, and as was shown in the 


Focal Lines of Cylindrical Lenses. 61 


previous paper, above referred to, the point of intersection /’ 
of the lines a’g' and c’d' is on the focal line; it is in fact 
the uppermost limit of the focal line corresponding to the 
semi-aperture /j. 

If ef’ = 1, and o’e’ =v we have from the triangle ¢’c'‘d’ 


Ep et = h( ="), a8) 


Drea iy) 


Now by the ordinary formula for spherical surfaces we 


have 
a RES 
ae Ce pani? pry —(p—l)oy 


and substituting this value of u in (1) we get 


—1 
but fe tr 
BY yi 


where 7; is the focal length of a spherical surface of radius 7,. 


a hiv 

1 re ’ 
or, the length of the axial focal line =axial aperture x distance 
of focal line from the surface x power of the surface. 


(b) The line at right angles to the axis of the cylinder; this 
may be called the tangential focal line. 


Two incident rays symmetrical with respect to the 
horizontal plane or the plane of fig. 2 are represented in 
elevation by the lines a’d' and ad’ and in plan by the line 
ad, and the corresponding refracted rays in elevation by the 
lines a'g’ and a9’ and in plan by the line ah. The two 
refracted rays will intersect in the horizontal plane in the 
point h which is at the extremity of the tangential focal line 
corresponding to the semi-aperture hy. 


If hg gies 1, amd og. = v3; 


we have from the triangles obe and ehg (fig. 2), 


Nan aks eg b= (2), . sie) 


hy te Vy 


(Ugh 


62 Mr. A. Whitwell on the Lengths of the 


But v, = wu and a Ld : = “(- = =). 


Us| u [iss Vy 
By eliminating u we find 


LM V2 
ve(u—1) Fen 


and substituting this value of v, in (2) we get 


— 


vy 


that is the length of the tangential focal line 
= the tangential aperture x distance of the line from the 
surface x power of the surface. 


2. To find the lengths of the focal lines of a 
sphero-cylindrical lens. 


Figs. 3 and 4 represent an elevation and a plan of the 
incident and refracted rays at the second or spherical surface, 
the corresponding rays at the first or cylindrical surface 
being represented in figs. 1 and 2. 


Let qq’ be the centre of the spherical surface the radius 
of which = —7,. 


(a) The anal focal line. 


The two symmetrical rays, represented in elevation by the 
line a//’, which were the refracted rays at the cylindrical 
surface, are now the incident rays. After being refracted at 


Focal Lines of Cylindrical Lenses. 63 


the spherical surface they will intersect at the point n! on 
the line joining the point 7’ to the centre gq’ of the spherical 
surface. The point n! is at the extremity of the axial 
focal line. 

If mn'=1; and o'm!' =v; we have from the triangles 


freq’ and n'm'q' (fig. 3), 


V3—1 U3 — Ts 
—— i e or iB = lL, cre e . ° (3) 
by eR Pir b2 


os 
iJ) 


From the ordinary formula for spherical lenses 


bal ac?) 
Or ee ea gts 


poe Brovs 
(= DOs Se De! 


we have 


and substituting this value of v, in (3) we get 


ls — n(*—) U3, 


or the length of the axial focal line of a sphero-cylindrical 
lens = axial aperture x distance of the line from the lens 
x the glass to air power of the cylindrical surface. 


Ee 
y 
or the power of a plano-cylindrical lens having a 
cylindrical surface of radius r,; or the difference of 

the two principal powers of the sphero-cylindrical lens. 


a the glass to air power of the cylindrical surface ; 


(b) Lhe tangential socal line. 


Two refracted rays symmetrical with respect to the hori- 
zontal plane or the plane of fig. 4 are represented in elevation 
by the lines a’n'p', and ap’, and in plan by the line amp. 
These two rays will intersect at a point p in the horizontal 
plane, and this point will be at the extremity of the 
tangential focal line pp corresponding to the aperture hy. If 
the semi-length of this line=/, and its distance trom o=1y, 
we have from fig. 4, 

1 ROU SE AL ag 
ie V3 U3 


(4) 


64 Mr. A. Whitwell on the Lengths of the 


epee 3: ii i 
ae = and —-— t= —D(2——) 
1 | ee sted ( ¢e (2=*) 
U3 V4 dat 3 "7 
or L, = ia( Hey: 
ry) 


the length of the tangential focal line of a sphero- 
cylindrical lens = the tangential aperture x distance of 
the line from the lens x glass to air power of the 
cylindrical surface. 
The results in 2 of course apply to a plano-cylindrical lens. 


3. To find the lengths of the focal lines of two plano-cylindrical 
lenses in contact, the axes being crossed at right angles. 


Fig. 5 is an end elevation of the system of rays looking 
in the direction of propagation of the light, that is from 
left to right, and fig. 6 is a front elevation. The axis of the 
first lens is vertical and its focal length = /;. The axis of 
the second lens is horizontal and its focal length = fo. 
The semi-apertures are h, and h, as before. Two rays, 
symmetrical with respect to the plane of fig. 6, incident at 
a’'b’ after refraction by the first lens will intersect at the 


point c’ at a distance from the lenses = o/v' = v, and at a 
hyv; 
Bi 


The second lens will bend these rays downwards to the 


distance from the axis of # = ¢'v' = 


position ad’, b'd'. The deviation c’d' will be =" since 


2 
7 is the deviation per unit of length along the axis of 2. 
2 


ER ee 


Focal Lines of Cylindrical Lenses. 65 


(ay The arial focal line ; that is the line parallel to the axis 
of the first lens. 

If 1, be the semi-length of the axial focal line, 

eo A 

ase va Fo 
Sel es) ee ener e 5 
1 1a hh iC (5) 

or the length of the axial focal line of two plano-cvlindrical 

lenses in: contact with the axes crossed at 90° = the axial 


aperture x distance of the line from the lenses x difference 
of the powers of the lenses. 


P= a t= OU —em = 


(b) The tangential focal line. 


If the line a’d' (fig. 5) be produced to meet the axis of x 
in e’ and if the line a’d’ (fig. 6) be produced to meet the axis 
of x in v’, /, = o'e’ is the semi-length of the tangential focal 
line and o’v? = v, is its distance from the lenses. 

From fig. 5 we have 

ls hy 


Lo hy—l’ 


and from fig. 6 we have 


hi-ho 


hy ty 
fi he ee ey HL ee 
b= hp p= bap = hora = i =) 
If f,=fo both and ,=0, or two plano-cylindrical 
lenses of the same focal length crossed at right angles are 
equivalent to a spherical lens. 


4. To find the lengths of the focal lines of two plano-cylindrical 
lenses in contact of focal lengths f, f, with axes crossed at 
an angle @. 
(a) First we will take the incident light to be parallel. 
(1) The azial line or that within the angle @. 


Fig. 7 is an end view of the system looking from left to 
right of fig. 8 which is a front elevation. 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. a 


66 Mr. A. Whitwell on the Lengths of the 


The line oc making an angle 0, with the vertical line oo’ 
(fig. 7) represents the axis of the first lens of which the focal 


0 = 
in: A pg Foleo 


length is 7;. The line oh making an angle @, with the line 
oo! represents the axis of the second lens the focal length of 
which is fo. Let a, b (fig. 7) be the points of incidence of 
two horizontal parallel rays symmetrical with respect to the 
line oo’. The axial semi-aperture oo’ = h,; the tangential 
semi-aperture o!a = o'b = ho. 

Consider first the ray incident at a. The first lens alone 
will deflect the ray ina plane at right angles to the axis f.. 
The line ac (fig. 7) will represent the emergent ray, the 
point ¢ being at a perpendicular distance f, from the lens. 
The second lens alone will deflect the ray in a plane at right 
angles to the axis /. The line ad (fig. 7) will represent the 
emergent ray, the point d being at a perpendicular distance 


fy from the lens. In the line ad take a point e such that 


Then the lines ac, ae, will represent the deviations in the 
focal plane of the first lens due to the two lenses separately 
and the diagonal af of the parallelogram ace will represent 
the deviation due to both lenses acting together *. 

The line af also represents in end view the ray after it 
has emerged from both lenses, and the line o/ (fig. 8) will 
represent the ray in front elevation ; of; (fig. 8) being made 
=f; The line ff; (fig. 8) represents the trace of the focal 
plane of the first lens. 

Consider secondly the ray incident at 4. The first lens 
alone will deflect the ray to a point g distant 7, from the 
lens; the second lens alone will deflect the ray to the 
point h distant f, from the lens. 

In the line b/ take a point 7 such that mah Then as 

2 

* See Chapter X. Handbook of Optics, by W. N. Suter. Macmillan 

& Co., 1899. ) 


Focal Lines of Cylindrical Lenses. 67 


before the diagonal j of the parallelogram bij will represent 
in end view the ray after emergence from both lenses whilst 
the line o’f (fig. 8) will represent the ray in front view. 

We shall now find what must be the relation between the 
angles 6, and 6, when the two emergent rays af and by 
intersect on the line oo’ or intersect in the central plane of 
which oo’ is the trace in fig. 7 or the plane of the paper 


in fig. 8. 
We have from fig. 7, 
ac=>= hy sin 6,—hzy cos @;, 


ae Lie — A (hy sin 0,+ hy COs G,), 
he J2 


bg = hysin @,+h, cos 6, 


bi — hi bh ms (hy sin 0, —h, COs @2). 
To Te 


From these values we can readily obtain the components 
of the diagonals a7 and bj resolved along and at right angles 
to the line 00’. 


_ Let the resolved component of af along 00’ = k. 
ajav G0" to oo =. 
bj along oo’ = m. 


bj at, 90° to oo = a. 


99 99 99 


Then 


fh, sin” Ghani 6, is O,+ B (h, sin? 8, + hy sin 8, cos 8), 
= i sin 8, cos 0; + hy cos? 8, + B (hy sin 6, cos 6 + hy cos? Oy), 

m=h, sin? 6, +h, sin 0; cos 6; + - (A; sin? 8, —/y sin 8, cos 6), 
n=h, sin 0, cos 6, i hy abe G, == (A; sin @ cos 8,— hy cos? 8... 


Now, if k=m, l=n, and kh? + ?=m’? +7’, the two diagonals 
will be equal in length and equally inclined to the line oo’, 
and as the pvints ab are symmetrical with respect to the 
line oo’, the two diagonals will intersect at some point p on 
this line. | 


F2. 


poteas 
aa 5 


68 Mr. A. Whitwell on the Lengths of the 
By equating k& and m or J and n» we get the relation 
71 Sin 8, cos 8,==/, sin 0; cos 6), 
and it will be found that this relation also satisfies the 
equation k? + [?=m?+n?. 

If this relation between @, and @, hold, then every pair of 
rays incident at symmetrical points such as a and 6 will, after 
refraction by both lenses, intersect in the central plane at 
some point on a line represented in fig. 7 by op, and in fig. 8 
by pq. This line pq is therefore the principal focal line. 


If we call its semi-length /,. 
From fig. 7 we have |; =op=h,— o'p. 


Now ee is & h=h~he;. 

Substituting the known values of & and J with the condition 
jf; sin @ cos 6,=/2 sin 8, cos 6, 

and simplifying, we get 

Fi cos 2024+ fo cos 24, 

Fi cos? 6; + fx cos By 


l, =f, 


If I’, be the principal axial focal length of the combination, 
we have 


l i. ie fs 
ants tiles otk 


and substituting the value of / with the condition 


— — ————— ee 


fi sin 2 cos 0.=f2 sin 4; cos 6, 
we have 


Sif i 


1 fi C08? 02+ f2 008? 6; cos? 0, _ 008? 8,” 
So Ii 
20 cos 20 
L=h,F adele” ue “ 
i aa ii 


(2) The tangential line. 


and 


__ If we produce the line af (fig. 7) to meet the axis of winx, 
then or will be the semi-length of the tangential focal 
line. If or=/, we have from fig. 7, 

ts ORS ga l 

Lap Voge Tame: 


f 
uy 
sy 
Se 
a* 
Ae 
cat 
i 
‘ 
res, 
ete 
ia 
v of 


Focal Lines of Cylindrical Lenses. 69 
and substituting the values of / and k we get 


fy, cos 2054 fy cos 20, 
Q_=— i, = . 7 > ° . 
‘ fo sin? 0, +f, sin? 6, 


If F, be the principal tangential focal length of the 
combination, we have from fig. 8, 


ee h 1 
Bag OP vat re v ; Fate sim? OG, | sin= 6," 
yay fe 
in cos 20, . cos 20, 
Pa ouanae ie Ala) 
Now 1, § €os7 8, coe? @, 
ee Ty 
hy Sin? Cees citieay 
Hg eee 
Ft jcos. 20, cos 20, 
ae? ee 
aN L=A,F, Gr x) ’ 


(b) Secondly, we will take the light diverging from a point 
at a distance —u from the point o (fig. 10). 


(1) Phe axial focal line. 


Figs. 9 and 10 are end and front elevations respectively, 
being similar views to figs. 7 and 8, the letters f,, fo, hy, ho, 
@,, @, having the same meaning as before. Consider first a 


pie eae 


ol ee ae 


AAO a 
cs iG a 


ray incident at a. This ray, after emerging from the first 
lens, will be represented in end view by the line as (fig. 9), 


70 Mr. A. Whitwell on the Lengths of the 
where os= —oc, and in front view by the line o's’ (fig. 10). 


] e 
The distance from the lens of the point s=ovj=v,. The 
point x is conjugate to the point —w with respect to the 
first lens. The ray as in passing through the second lens is 


bent towards the axis of the second lens, the deviation sf or 


3 é : ad : ad 
ae corresponding to the distance v, being = 4», since rs 
2 


Q 
is the deviation per unit length along the axis of w, this 
deviation being measured along a line at right angles to the 
axis od. The line af will therefore represent in end view 
the ray after emerging from both lenses. The emergent 
ray is represented in front elevation by the line o'fv; in fig. 10. 

Now vy r 
0s= oe Fa (h; cos 0, + ho sin 6,), 


(ea (hy sin 0) + hy cos 02). 
So te 
From these values we can get as before the resolved 
components of the diagonal af along the line oo!=k and at 
right angles to this line =1. 


k= 7 (ly sin? 0, + hy sin 0, cos 02) “4 (hy cos? 0; + hy sin 6, cos 6;) +hy, 
2 1 


a (hz cos? 6, +h, sin 8, cos 8) fe (hg sin? , +h; sin 6, cos 6;) + hg. 
; i 


The corresponding values m and n for the resolved com- 
ponents of the line bj, obtained by considering a symmetrical 
ray incident at b and making a similar construction, are 


mao (h, sin? @.—hg sin 8 cos 5) — Fa (hy cos? @,—h,y sin 8; cos O;) + hy, 
n= 5 (hz cos? 8, —h, sin 6, cos 6,) — Zz (hy sin? 0; + hy sin 6, cos 6,) +h. 


By equating & to m and / to n we get the condition under 
which the lines af and 67 intersect on the line 00’, viz., 


J, sin @, cos 0. —f2 sin 0, cos 0; =0.. . . (6) 
If the semi-length of the axial focal line =J,, from 
figs. 9 and 10 we have 
h,—l, __k 
| ee — i 5 * e ry e ¢ ° (7) 


Or Fatty 


k 


h=h—-5 Ros } 


Focal Lines of Cylindrical Lenses. 12 


If the distance of the point p or v, (fig. 10) from the 
point o or from the lenses be called v,, we have from fig. 10 
Gy ay ok but from fig. 9 =- f 


hy—l, ‘ lis 


ou hy—t ; 
nd 


(3) 


From equations 7 and 8, by substituting the values of 
k and J with the condition (6), we get 


20 si2 
h=hn( 2 2 eS ="), 


ah, lla 
or L=hw( 9 — a), 


Ug hy i 


an analogous result to that obtained for parallel light. 


(2) The tangential focal line. 


Let the line af be produced to meet the axis of w at r (fig. 9) 
and the line o'f be produced to meet the axis of z in the 
point v; (fig. 10). The semi-length of the tangential focal 
line =or=l,, and the distance of the line from the lenses 
=ov,;=v3. From fig. 9 we have 


Ly +} h l 
2 / ee or Ip =hyz —he, ~ ° : (9) 
and from fig. 10 we have 
Ua = k e e . e ° e © ° (10) 
3 h, 


From equations 9 and 10, by substituting the values of 
k and / and putting in the condition (6), we get 


cos 26, ‘ cos 20, 


bah (MS + a): 
ih 1 
=tohe(g, 7) 


5. The lengths of the focal lines of a number of cylindrical 
lenses in contact arranged with their axes crossed at any 
angles. 

The results obtained in the last section are perfectly 
general. Let the focal lengths of the lenses be /,, J2, Tee, 
and let the angles between the axes of the lenses and any 
fixed line such as oo', figs. 7 and 9, be 01, 42, 45, &c., the 


72 Mr. A. Whitwell on the Lengths of the 


angles being reckoned positive in one direction and negative 
in the other. The condition that must be fulfilled if two 
parallel rays symmetrical with respect to the plane con- 
taining the line oo’ and the optic axis are to be refracted 
so as to intersect in this plane, 1s 


in 6 cos @ 
s sin 
y 


The principal focal lengths of the system are 


=(), 


af 
FF = —.\}. 
: < cos? 8’ 
Ng 
F,= Ene" 
"maa sin? @ 


and the semi-lengths of the focal lines are 


2 9 
L=h Fi 20 =h,F, (F =F 
] 


f —*#,; 
7 Wes COS 20 _ G-- 4) 
le hol Qa f —_ His Ey HR 


6. To find the lengths ef the two focal lines of two plano- 
cylindrical lenses of focal lengths f, and fp with axes 
crossed at 90°, the distance between the lenses being =6. 


Figs. 11 and 12 are elevation and plan respectively of the 
system of rays. 


(a) The line paraliel to the axis of the first lens which is 
vertical. 


Two rays diverging from a point uw’, at a distance —u 
from the first lens, and incident thereon at symmetrical 
points a, b, a’, b', will, after refraction by the first lens, 
intersect in the point ¢ (fig. 11) on the line represented in 
elevation by c’v,;' (fig. 11), and in plan by the point v, in 
fig./12, | 

Let Ser = Vj and FPR ee ne 

Then we have 


The two refracted rays on meeting the second lens in 
points distant hs from the axis of # are bent downwards, the — 
point of intersection being at d’, the distance c’d’ or the 


Focal Lines of Cylindrical Lenses. 73 


deviation due to the second lens being = is (v,—6), since : 
2 


is the deviation per unit length along the fav fivy, and vy—8 
is the distance pv)’. 


Let 1, be the semi-length of the line. 
Prom fig. 11 we have 


= —B(e-8) Seeeen ara ss 
and hyv, ] 
hg—hy fi ; 
5 = ? 


V7} 
Eee 
ig=hy 4 8(= — ith. 
: ny Jy Je 
Substituting this value of h; in (11) and simplifying, 
we get 
6 6? EZ 
bah fo ( 5-5 — _ pene ee 
: cp fi i fife) © fhe Sars 


= fps, (MLefi—8) + (8 +24) BY; } 


If v,;=f,, or the incident line is parallel, 


1 Na } 
=h See . 
hi E FR) 
(b) The line parallel to the axis of the second lens which 
2s horizontal. 
Produce the refracted ray h3d’ (fig. 11) to meet the 


horizontal plane in the point », 7!. 


Then v,¢==/, is the semi-length of the focal line. 


74 Mr. A. Whitwell on the Lengths of the 


From fig. 12 we have 


or h 
— zee Vo + 6) —he ° ° ° ° ° (12) 
vy 
Now ae aes cl 
— - + v1 Tit . e ° e e ° (13) 
and 1 eee 


SS ae . 0 
1 _ 8t2—8fe— Vofot vofi-hihr 
tay Sv.f, — of te ae ie 
and substituting this value in (12) we get | 
lp _ fyi vs" (fo—fi — 9) + ¥2(26f, — =) Oa 
trite Hd —Svof; 
If wm=fo, 


1 é 
L=lefi(-—7 +47). 
Mell CT Gs 
From fig. 11 it will be seen that when c/d'=c’'v,', we shall 
have v;=t,4+ 6 and the lengths of the focal lines =0. 


This will happen when the points conjugate to —w with 
respect to the two lenses separately coincide, or when 


vy (fo—fi—8) + 01 (6? + 26f;) —P fi =0 
The form of this equation shows that when f, and fj are 
constant there are two values of v, corresponding to each 


value of 6 which will make the lengths of the focal lines 
ae 


7. To find the lengths of the focal lines of a _ sphero- 
cylindrical lens when the source of light is not on the 
optic axis. 


(a) The axial focal line. 


Fig. 13 isafront elevation of the system. We can regard 
tre lens as made up of a plano-cylindrical lens of focal 
length f; and of a plano-spherical Jens of focal length fy. 

i 
+5, the 


| latter of which we will call . Let a be he pene” of light 


The two powers of the combination are é and ++ 


situated at a distance -au = a from the axis ow-and at a 


Focal Lines of Cylindrical Lenses. 75 


distance ow=—uw from the lens, and let the semi-aperture 
=h,. Consider first the Barnetion by the plano-cylindrical 


lens alone. The part of the lens above the line ad will 
produce a focal line be of length (ha) 5 at a distance 
ov,=, from the lens. The part of the lens below the line 
ab will produce a focal ot bd of length (h, + “) The 
total length of the line is Oh, 2 


Consider now the rebhe ee a the plano-spherical lens. 
A pair of rays, symmetrical with respect to the plane of the 
paper in fig. 13, converging to the point ¢, fig. 13, will be 
refracted in such a way that their point of intersection 
will be at e on the line oc, the distance of e from the 
lens being =ov, or v,. Similarly a pair of rays converging 
to d will, ufter refraction by the spherical lens, intersect at y) 
on the line od. The axial focal line will therefore be ef. If 
its total length =21,, we have from fig. 13 


= Iw, 
a ° ater ; 
that is the length of the line is the same as when the point a 
or the source of light is on the optic axis ou. 
If the length of that part of the line above the axis ou be 
called y we have 


(h, —a) 5 +a 
as cated 


UD vy 


eo) eee EN ah Noe 
Sen tele eye \ oe 
he” ates aie fa ce 


76 Mr. A. Whitwell on the Lengths of the 


but eo ei or dis a 
Ges. fs daa aaa 
U=— Uy oe Tareas 15 
d 7," ii (15) 


This is the equation of a line which forms the upper limit 
of the focal line as its distance from the lens varies. It is 
represented in fig. 14 by the line ag. The intercept on 


the axis of # = og = ABIL ED and the intercept on the axis 
VG fags eee 
of y=oa=a. When »=fs, y o hy. 
dis: 
If in equation 15 we put h,=—h, we get the equation of 


the line ak, fig. 14, which forms the lower limit of the focal 


line. The intercept on the axis of y=a and that on the axis 


OE as pe ae 
te = ok = he tafe When »,=/3, y= 7 hy. 
If in equation 15 we put a=—a and h;=h, and —h, we 
get the equations of the two lines —ag and —ak which 
form the limits of the focal lines when the source is at —a. 
Fig. 14 shows that if we have two sources of light at a 
distance 2a apart the axial focal lines will coincide at the 


principal focus, their lengths being = 2p, 78 and will overlap 
1 


if their distance from the lens lies between the values 


CANE fits 


og = «6and ok = 


apy — hy fs afi th fs 


Focal Lines of Cylindrical Lenses. 77 
(b) The tangential focal line. 


Fig. 15 shows a plan view of the system. A pair of rays 
symmetrical with respect to the plane of fig. 15 diverging 
from a point a, at a distance aju=a, from the axis ou and at 


a distance ow=—vw from the lens, and incident at a point hz 
at a distance ohy=h,. from the axis ow will, after refraction 
by both surfaces, intersect in the plane of the paper fig. 15, 
at a point 6 the distance of which from the lens =ov3=2;. 
A similar pair of rays incident at the corresponding point 
—h, will intersect at cand be will be the tangential focal 
line. Let the length be=2/. 


Now 
eae 
U Vg is 
ji aN it 


Subtracting one equation from the other we get 
Ee INL sil SCO ee a 
Be te hg! yf re eet an 


From fig. 15 we have 


ly 


-~ 


pores 
Be bof th 


pe Ushy 
Dias re ? 
that is the length of the tangential focal line is the same as 
when the source of light is on the axis ou. 
The equations to the two lines which form the upper and 


lower limits of the tangential focal line when its distance 
from the lens varies are 


i= Wye rio be. 


Now 
— Use ay QAyV 
a SS or) we 
Us Sh U 
AyV3 l J Vshq 
LS +l, — as (— AUN ATS maa 
Uu US) Gig hi 
or 


yrs , Ugh, 
y = ay — rai 1 ie 
een 


These lines are shown on fig. 16 by ng and nk. 


When a, is made = — a, we get the equations of lines mg, 
mk (fig. 16). 
The intercepts on the axis of y are +a, and those on the 
axis of w are 
hfe 
a frths/s 
Wig. 16 shows that if we have two sources of light at a 
distance 2a apart the tangential focal lines will be distinct if 
their distance from the lens is less than 


fay ee Lah hy 
ay fi ths fo 
or greater than 


D0 [eS 2 es 


but will overlap if they are formed anywhere between these 
two distances. 


8. The image produced by a sphero-cylindrical lens of an 
object consisting of narrow parallel horizontal and vertical 


bands or slits of light. 


(1) The images of the horizontal bands. 


Let the distance between the centres of the object bands 
be a and the width of the bands be w. Each point of each 
horizontal band will produce an axial and a tangential focal 


Focal Lines of Cylindrical Lenses. 79 


line. At the tangential focus all the tangential focal lines 
due to one of the object bands will together form a horizontal 
image band the width of which parallel to the axis of the 


cylindrical surface of the lens will be I and the distance 


. e Av: 
between the centres of the image bands will be a where 


v is the distance of the image and —vw the distance of the 
object from the lens. 

Consider now what will happen at the axial focus. Every 
point in one of the horizontal object bands will produce an 
2hyvo 
a 
distance of the image from the lens h, the axial semi- 
aperture, and /, the focal length of the plano-cylindrical part 
of the lens. ‘The horizontal image bands will have a width 
parallel to the axis of the cylindrical surface of the lens 


axial focal line of which the length = , where v, is the 


the first term being the width which the image band would 
have if the lens were spherical. ‘The distance between the 


Fig. 17 (p. 80) is a 


similar view to fig. 14 but showing the limits of the focal lines 
due to five narrow object bands, the distance apart being =a 
and the width w being small enough to be neglected. The 
lines A, B, C, D are drawn at distances from the lens by 
putting a=1, 2, 3, or 4 in the formula 


2 av 
centres of the image bands will be —”. 
2 —uU 


and the lines I, H, G, F at distances obtained by making 
a=1, 2, 3, or 4 in the formula 


Oi auth Ja yy 
af + 2f3hy 


If the object be placed at such a distance from the lens 
ihat the image is formed at the line A or I, the edges of 
adjacent image bands will coincide and there will appear to 
be no image at all; a screen placed at A or I will be very 
nearly uniformly illuminated. If the images are formed 
nearer to the lens than I or further away than A, five 
separate and distinct image bands will be formed. Again, if 


80 Mr. A. Whitwell on the Lengths of the 


the images be formed at the lines B or H it will be seen that 
the edges of the image bands due to the object bands 1, 3 
and 2, 4 and 3, 5 coincide, and we shall get the central part 


of the screen almost uniformly illuminated from two slits 
whilst the top and bottom will be illuminated from one slit 
only. Again, if the images are formed at C or G the edges 
of the image bands due to slits 1, 4 and 2, 5 coincide and 
we get the centre of the screen illuminated by three slits, the 
outer parts being illuminated from two slits and the edges 
from one slit only. Similarly, if the images are formed at D 
or F the edges of the image bands due to slits 1 and 5 
coincide and the centre of the screen is illuminated from 
four slits, the light fading off towards the top and bottom 
edges as the screen is illuminated from three, two, and one 
slit only. When the object is at a great distance all the 
images coincide at F the principal axial focus. 


Focal Lines of Cylindrical Lenses. 81 


The table, fig. 18, shows the number and amount of 
illumination of the image bands due to five object bands, the 


letters representing the same lines as in fig. 17 and the 
numerals the number of slits which illuminate the particular 
image bands; for instance, if the image bands are formed at 
some point between the lines B and C there will appear to be 
three bands illuminated from three slits separated by bands 
illuminated from two slits, the edges of the screen being 
illuminated from one slit. 


(2) The wmages of the vertical slits. 


Each point of each vertical object band or slit will form 
an axial and a tangential focal line. At the axial focus all 
the axial focal lines due to the points on one vertical object 
band will together form one vertical image band, and if 
—u and v2 be the distances of object and image from the 
lens the distance between the vertical image bands will be 


== and their width in a horizontal direction will be 


= Consider now what will happen at the tangential 
focus. Every. point on one of the vertical object bands will 
form a tangential focal line of which the length will be 
ae Zhovs 


, v3 being the distance of the line from the lens, h, 


1 
Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. G 


R2 Mr. E. Howard Smart on a Formula for 


the tangential semi-aperture, and /, the focal length of the 
lano-cylindrical component of the lens. The vertical image 
fo de formed at the tangential focus will have a horizontal 


width 


wr; , 2hots 


ut 


the first term being the width the bands would have were 
the lens spherical. The distances between the centres of the 


vertical image bands at the tangential focus will be =. 


If fig. 17 be regarded as a plan view analogous to fig. 16 
instead of an elevation analogous to fig. 14, then all that has 
been said about the images of the horizontal bands at the 
axial focus will apply to the images of the vertical bands at 
the tangential focus. The lines A to I are, however, to be 
obtained by putting a=1, 2, 3, or 4 in the formula 


Gh Te iZ 
afi t+ 2he fo 


V. A Formula for the Spherical Aberration in a Lens- 
System correct to the Fourth Power of the Aperture. By 
K. Howarp Smart, M.A., Head of Mathematical Depart- 
ment, Birkbeck College *. 


ic . ordinary formule as given in the text-books for 

central spherical aberration are computed to the square 
of the aperture only,—a degree of approximation which is 
insufficient for the purpose of the practical optician in the 
design of photographic and other objectives. In the 
following work a formula will be given for the longitudinal 
aberration for a system of coaxial spherical surfaces separating 
media of refractive indices py...f. which is correct to the 
fourth power of the aperture. A greater degree of accuracy 
than this is usually undesirable, the complexity of the 
additional corrections being out of all proportion to their 
usefulness. 

Let the spherical surface AP of radius 7; separate media 
of refractive indices y;_; and p;. Let the ray OP be incident 


* Communicated by the Author. 


the Spherical Aberration in a Lens-System. 83 


at a distance y; from the axis. If s; and s,/ denote the 
distances (measured from the vertex positively to the right) 
of the intermediate point images formed by paraxial rays 


respectively before and after refraction, and A;_,, A; the 
corresponding spherical aberrations for the rays OP, PI, we 
have AO=s;+A;_1, Al=s/+A;. Then C being the centre 
of the surface AP we have as usual 


fy—1 SIN? = pe, sin 7’, 
whence accurately 
Hi-1 + CO te piCl 
eon | TRS. 
pa-1(AO— 7%) OC, 
/A0?—2(AO—r:)AN  ./AP?—2(AT—r) AN* 
Let AN =X, then 
Mi-1(s; + Ai_1 — 7%) = pi(s/ +A;—7;) 
WAC? + Ae)? —2X(8;+ A;-1 = Ps) A/a + A;)?—2X(s;/+ A; = r;) 
Expanding each side- in powers of the A’s by Taylor’s 
theorem we have 


Hint { #8) + Aj_if’(s,/) + = Ae fi). i 


= wd ADAP) + FARA). b, 
G 2 | 


s4 Mr. E. Howard Smart on a Formula for 
where 
a Se’ A 
f(s) = anlar 
( ) a/ 82 — 2X(s;—7%) 


f'(s) = B= 


ay 7 (5s;:— 27;) + 2X(s? — ris; +7) — 2 
{si of2 2(si—7 i) }? ? 


with similar expressions for the dashed letters. 
Expanding these in powers of \ 


; 5 2 3 a vr; : 
f(s) = (1-") +> (1-8) +3 all- ao ff 
Since NO (#) 4 5(4 ) 

"; oe ; 5 vi 


to the fourth power of (y/r), 


fie) = (1-7) + 5-0-2) G) 
pig tee mG = Us ) Cok 
rivet (1-2-2) (8) 
ae - J-G-0)- S 
“LAHEY 
ae aye —)(vs a n=) s9(2))} 
BGs fs) 
)(-25 sane a 
am ieee) 
—3(%*) (149%|— 68m) “30% ) ~25%)} 


these correct to the fourth power of a 
ie 


t 


f"'(s)=— 


the Spherical Aberration in a Lens-System. 85 


: _ The expansion therefore takes the form 
a * aie : 
nf {1—") +3 (G2) (HY + E1- 2) (80-2) wy 
a Sz 8; Si VT; 
| rei, § ( — =) a 1 —2)(1- Vi 
Bee fi Hn (is 8) 112 )(1438~2(2) 
YT; 3 Yi 4 
sn fs 2(1-80(1-2)) (6) (40 (2) 02) 
ae {1-5(1- : ‘)) i 3(143 Si BS S; 
Ja CNY 
+80(F) -25("")) = 


=a similar expression in mi, Aj, and s/. 
aM 2 : yi \? 
To a first approximation neglecting and A we get 
4 


of course 
att =) = ik 
scm ( Sj wi(1—2 a Si 


Let each of these be denoted by Qs,. 


To a second approximation 


LD) 


I 1 “A; 
: a (1- ") uy a ee f. 


2S; 

Whence 
pil: ne fbn ey ees aye (— -—) 
Si si Pinr8i pi!) 


And if 6; be the inclination of the ray to the axis in the 

medium whose refractive index is pi: 
G= 153 — ise ee 

to this degree of accuracy, and therefore summing the series 
of difference equations (as in H. T. Whittaker’s tract on 
the *‘ Theory of Optical Instruments ’) 

ho Ua ee 1 1 ) 

BE ed BA Medes 

prArG;. ra 2 = Qu (; [Syl 


p-1Sp 


! 
Se Mpn—1Sp pS 


Aart 12=* i i 
“n= Be Uae -—.,). 


86 Mr. E. Howard Smart on a Formula for 
Proceeding to a third approximation we may assume A as 


of the order Ay? and neglecting powers of = above the 
fourth, 


SCN) “STO DE) 
FP 1-10-2)0-)C)) 


=a Ane expression in #4, Aj, and s,. 


We can use the relation @:-1s; = 0;s,' = y; again correct to 
this order of approximation. 
aoe 
A ea 
piAidi pees -1 sah “O (—_- pisil 


ly ( ] ane 6 i 1 
o 8 rz Qs: peas * ue 7 aes Oe 1382 wes; 12 


1 Yi eee 1— las 1 (1- “:) a-= a ar; (1-2 fi (1- =) 
sf? Sj 


fe 172 5 


12 12 
BAS | Peas 
+y7 ats — eae 


where A’ denotes the value of A as obtained from the last 


approximation. 
Summing the difference equations for the system of 


refracting surfaces and putting ©; for 


i 
Gases aeeal 


we - 
2 4 “Yr 1 =) 
[Ax ie 2 a3 Yi O45 oe Ort § a oi 5:8 ‘laa au psi’ 


a= 
LSS Yi A pi A alee Bs Sue yi (A! ba A’;- os 


sa elas, 2 
Fie N iS; 7 8,” ae v% ye 8; 


bo 


I=) 
em fa ; Wy oe . i=k A! ae A’?,_ 
te ae(Gr 2h) + Sells 


le 


the Spherical Aberration in a Lens-System. 87 
Substituting for 
A! i fe. AY ihes 


S42 $7 


i Pe 2 ( 1 = 
5 YE Ws, aaa Ee 


or 2 y®® before obtained, the expression takes the final 


the value 


shape 
ae. = FE ytOits Ses E H/0,0 — Sage 
Mx oe i 2 (Wipes bee iS; 
a Al v 7S a i vey aAG 4 ) 1— 
m1 $2 (Sips) 25 g(a) 
i=1 °* : 
Jel FANS z DO stn Veal 
+ 3y 2 (Be 1 5 i) 
or 4S, = a x above expression. 
Mr x 
(the last five terms being required by the higher order of 
approximation). 


To test this let us apply it to the case of a single thin lens 
of radii r and s in air, and let wu, s, v be the distances of the 
“a and its successive images from the lens. We have 


Pe eagle eee Deke 
tee 9 y®,, 


B 
oe te) (iy ie VC - a tS 5 
Q==-F=n(-- 2), e, =(- Re a 
ie ait 
AE ae Th ay (@,+90,). 

Also meine at p—l 

Eh ApS,“ fb pr” 

4s fis wWw+l pl 

[Sy hee cE ut a 


88 Mr. E. Howard Smart on a Formula for 


The formula reduces in this case to 


[5 @.t@yye4 o(a+ 5 J) + (0 (5+ = 


ray 


] ara ee u ee oid 
+0,6,(-— +1))u-2 ee + i apr : y 


Ors Se 


3 NS yr 2 
ale: aes 3 
a7 A i 
or 


iN ae 1 l\(/et+1 , wD | 
Horsenres ive[(0+ 9) +2 (o-) Ee) 


Ss” uU U 7 


aol 74] —1 0,/1 1 1 
Ga fa Hot) has <(=-5) e \ 
r C. F v ui 3 ‘ SN os 4 oe 


6 
—2(G4 e,) | i : y'v(@, + ®,). 


The first term gives the previous approximation. 


Dealing 
with the second the coefficient of ave v? is 
3 p—l eg! 
ofhe20-Der'+#7)-20-D} 
$] 
3 — 4 By me 
40, +3() aye a *) } 3 —2(@,+ @,) (= — 3) 
Si — se 1 
ince Bb _ Fb Le ne 
$y , § v0 


m  t-=C-D-C-1) 


this may be written 


©; eames 3. be OT | 
pe vid ee 


TU 


@,f wW—-wt+3  By?t+tu—6 3(W?—1 
+o ee a 


SU 


2 ss ish say Ap? = 3? 
ely eatcle 1a} Ma) 
s iy as ae sv v 
eee fit sls. ae man OA 


a Ag SU 


Piha 
pe 


the Spherical Aberration in a Lens-System. 89 
p—l 
. 


Substituting for = as above in terms of s and v 


the coefficients of 2@,/u? and 2@,/p? are easily seen to be 
equal and each may be written 


in) -DixG-*2)) 


The whole expression for the longitudinal aberration 


is thus 
] wheels aks Gets) Ge tes) seta) 
70+ Oye? + Za ye'[0.4 © 2 ) | TU Se ae 
pe (Si © 3(w2—1 
+0) Fo pte 5 fe = St 


+3ie—nor09 {aan (tf x(t-2)0-)} 


i 2 
+ 4 y*v* (Oy; == 0,)’, 


where ©, and ©, have the values given above. 
Some writers give the spherical aberration as a correction 


to 5 instead of v. Assuming this correction to be 
ace Eh), 


the longitudinal aberration for a thin lens is 
if(4 -ay—By)-» or (Ay?-+ By*)u? + A%y4v3 


to this order of approximation. Comparing with the above 
the result for a thin lens (allowing for the necessary changes 
in notation) is seen to agree with Messrs. Herman and Dennis 
Taylor’s formula as given in the latter’s ‘System of Applied ~ 
Cptics’ (Appendix, pp. 6 and 13). 
In applying the formula to any lens system a relation 
_between successive y’s is needed. 
If d; is the distance between the zth and (¢+1)th surfaces, 
P:N;, PisiNii1 the successive ordinates, A,, Aji; the vertices 
of the surfaces, it is easily seen that 


ee 
Yor yy GAN A Ninn ee as 
Ce s; +A,—A:N; 


to the 2nd power of the y’s. 


90 Mr. E. Howard Smart on a Formula for 
Expanding and remembering that A’; is of the order y;’ 
we get easily 


; fo) aN ie di 5 i; \? 
teary Ml 14%) ow 4 


Yi cea es 287% 258; Vi41 Si 


using the first approximation for y;,; 1n terms of y. . 
In our formula, therefore, it will be usually sufficient to 


substitute 
d; 
ne == ~) for Yi4+1> 


We shall conclude the paper by applying the formula to a 
thick double convex lens on which is incident a beam parallel 
to the axis. Let 7, 7, be the radii of the surfaces. Then 
s;=0, and it must be remembered that successive s’s are 
measured from the vertices of the respective surfaces, and 
let sp’=v. The formula takes the form :— 


wri B} 6 Yoo 
pa Lg Wi +9:%0,) + 5(43 0+ 26.) 
3 iL 1 ir 1 
a 8 {9:°Q1®; ( a9 ==) + y2°QoO. (~ + . 
ya get 1 yo (A's BAY, =") } + su 
sy {2 vr} Se Ca Geass 2 yy 


pd’? AG A’? 
+{y2h B y +9 (= - Hr) h]. 


Take the following numerical values 


soe(Qh-*8)} 


$5 ge is ; t (the lens-thickness at the middle) ‘75, y=", 
then by calculation s'=—3, s=—2'25, sf=v=—1:2,  y,=°3, 


72 
Qi=- iL Q. = 7/6, 6,= 2/9, 0, = we uA’ =°16, 
ay Bigs 


2 = 5300! ©, +7@,) = 064,498. 


vw 


the Spherical Aberration in a Lens-System. 91 


The rest of the numerical work may be arranged as 
follows :— 


+ — 
L 4 pa e rs } 
2nd order | gf) es ae eas 
corrections 1 
5 92's = (002,960 
‘005,804 
3,6 
( 3H, = ‘000,341 
| 3 Yor = e OD) 
| 8 re ©, — 000,022 
anne. SS a 8 
gg HO hie) 0 "000,086 
5 
3 9 8Q,0:(— +7) = 000,353 
2 
4 ! : 
3rd order 2. fea = 000,303 
< 1 
corrections — 242 “(Gi ) = 000,234 
2 
ee 
2H ar = (00,076 
ys ag om = 000,489 
| 
Ue mar = ‘000,101 
S} 
i) 12 
. wAi-#2) = 000,082 
()] S9 
‘007,042  -000,839 
‘000,839 
+ 006,203 


To get the final are) of A, this must be multiplied by 
or 16. This gives +°099,248. 


An accurate tracing of the extreme ray through this lens 
by a trigonometrical method gave for the longitudinal 
aberration +°099,37, showing a negligible discrepancy of 
00012 approximately. 


Feb. 3, 1910, 


Na ee Ly #82 yc] 


VI. The Problem of Uniform Rotation treated on the Principle 
of Relativity, By G. Strap, B.A., and H. Donapson, 
B.Sc., Cavendish Laboratory, Cambridge *. 


| pened EST (Phys. Zeit. Nov. 1909, Science Abstracts, 
Jan. 1910) advances the problem of the rotation of 
a solid cylinder about its axis, in connexion with the Prin- 
ciple of Relativity. He suggests that a contradiction is 
involved from the facts that any element of circumference, 


which must be moving in the direction of its length, tends 


a 


to contract in the usual ratio NA 1—-:1, where c is the 


Cc? 
velocity of light and v the velocity of the element, whereas 
any radius tends to remain unaltered, because it moves in a 
direction perpendicular to its own length. A quantitative 
solution of the problem in the simpler case in which the 
rotating cylinder is reduced to a rotating disk has led to 
rather interesting conclusions, and is here given. 

Consider the disk rotating about an axis through its centre 
perpendicular to its plane. In a small sector AOB of angle 


68, any length ab, at a distance r from O, will contract from 


2 
r.80 to r.80, /1—- when the disk is rotating, so that 


a 
ab is moving with linear velocity v. 

As Ehrenfest pointed out, the Oa will have no tendency 
to change, and if this condition is to be fulfilled it is im- 
possible for the disk to remain in the plane form. It must 
assume a cup-like form, whose horizontal sections will, from 
symmetry, be circles, and whose shape is such that ab has 


Z 
contracted to an/ 1-— a while Oa is unaltered. 


* Communicated by the Authors. 


Problem of Uniform Rotation. 93 


If, therefore, A‘YOA represent the vertical section of the 
final form of the disk containing the axis of rotation OX, 


x 


we shall have Oa measured along the are equal to 7, while 


2 
aB measured perpendicular to OB will be r/ 1— =. In 


this way both the conditions demanded by the relativity 
principle will be satisfied. 

Writing Oa=s and aB=~y, according to the usual 
notation, we have 


: eC— ce -7'o" 
Y = = § SS 5 
U C2 lea 


w being the angular velocity of the disk. 


CE a SPCR eS, sii) 5) io a 5, Css) 
Differentiating, and arranging terms, we have 
(c? + w’s”) yo —s¢ (?—y’w") 
or (Cars iy Cos @ — SC ya), ve ees, (1) 


Substituting in (ii.) the value of y from (i.) we have, 
taking the positive root of the equation, 


2 02,2 
(2+ wis?y- SC COS h — ya ORS 
2 2.2 2 2 62 ? 

Vc +s C +-o’*s 


whence 
Ce E@IS PF COS N Cy bald 0) ai «1 (UIE) 


This gives the intrinsic equation of a section of the disk 
when rotating with angular velocity w, and contains no 
approximations. _ , . ; 


94 Problem of Uniform Rotation. 


Case I—When the velocity of any point on the disk is 
small compared with the velocity of light, we have 


9 
D) 


C 


2 .2\ 3/2 * 
WS 
ee? 


The conditions of this case will be satisfied if ws is small 
compared with c. 
Thus we may write 


cos b = 


2 2 
cos p = 1—3/2—5 f 
whence 
2 tp 
s=—.-s5§ lee 3 Le rr 
V3 we sin 9 (iv.) 


This indicates that the form of the vertical section in this 
case is a curve of the cycloid family, an epicycloid. 

Case IJ.—When the velocity of a point on the outer part 
of the disk approaches the velocity of light, since we have 


pct Sai 
0 e+ ors? 
and v= yo, 
we get 
SCw c 


> 4) BE wea ae ° ° ° (v.) 


From this we see that for all values of s which differ from 
zero by any finite quantity, v=c when @ is infinite. Thus 
no point on the disk can be made to move with a velocity 
greater than that of light, which is exactly what would be 
expected from relativity principles. 

Further, from the equation 


1 


if). bee 
(i+ “) 


we see that, when w becomes very large, cos¢ is small, and 
is also sensibly independent of s, unless s is very small. 
Hence, when the angular velocity becomes very large, the 
disk approaches the form of a right circular cone of small 
angle, except near the centre of the disk. 

When o is infinite, all points at a finite distance from the 
centre of the disk are at zero distance from the axis of 


cosh = 


Formula for the Discharge over a Broad-crested Weir. 95 


rotation, 2. é. the disk has become a straight line coinciding 
with the axis of rotation and of length equal to the original 
radius of the disk. 

This straight line, of infinite density, is analogous to the 
plane of infinite density obtained by moving a solid body in 
a straight line with the velocity of light. 

The difficulty of experimentally discriminating between 
this solution of the problem and the solution which considers 
the rotating disk as contracting but still remaining in one 
plane would be great. If we assume that light is reflected 
from a mirror fixed normally to the disk, and assume that we 
can detect a deflexion of 1 mm. in the position of the reflected 
beam received on a scale 10 metres distant, 2. e. a value of 
equal to a0? it will still be necessary to have a frequency 
of revolution of about 1000 per second to produce this effect. 

In conclusion, it would seem probable that, for a disk of 
any appreciable thickness, the plane position would be main- 
tained during the rotation, the material of the disk being 
strained, in which case Ehrenfest’s contention, that we have 
here a contraction of a line in a direction perpendicular to 
its direction of motion, is valid. On the other hand the 
above theory does away with this difficulty, but involves a 
change of form of the disk, which does not, however, lead 
to any conclusions not in perfect accordance with relativity 
principles. 


VII. A Rational Formula for the Discharge over a Broad- 


crested Weir. By Professor A. H. Gipson, D.Sc., 
University College, Dundee *. 


A was first pointed out by Dr. W. C. Unwin, an 


expression tor the flow over a broad-crested weir may 


be deduced from first principles if it be assumed that the 


crest is so wide in the direction of flow, that the water 
settles down betore leaving the crest, to form a parallel 
stream of thickness ¢, and that in this stream the pressure at 
any point is that corresponding to its depth. Thus, assuming 
the velocity in the surface, and at every point in this stream 
to be given by 29(H-—t) ft. per second, where H is the 
up-stream head measured ahove the crest, the discharge is 
given by Q= bt V2g(H—t) cub. ft. per second, where 5b is 
the breadth of the stream. 

As the stream will adjust itself so as to give maximum 
discharge under given conditions, ¢ can be determined by 


* Communicated by the Author, 


— 


96 Formula for the Discharge over a Broad-crested Weir. 
equating - to zero. This gives ¢=2H, and on substituting 


this value we get Q =°3850 /2¢.H® o.r.s. Writing this 
in the usual form, Q=2Cb/2qH? = KbH? we have 
= otocaut as =o 087. : 

This method of treatment becomes more rational if account 
be taken of the fact that in a parallel stream flowing in an 
open channel, the distribution of velocity over any vertical 
is not uniform, being a maximum at or near the surface and 
a minimum at the bottom. 

Experiments show that the ratio of the mean velocity over 
the section of such a stream, to the maximum surface velocity, 
while varying with the depth, width, and roughness of the 
bottom of the channel, lies between the limits ‘82 and °87 
for such surfaces and depths as are common on the 
crests of such weirs, this ratio increasing with the depth 
of water. 

Assuming, as is practically the case, that the maximum 
surface velocity in the case of the weir is equal to 4/2g(H—+2), 
the mean velocity will equal & /29(H—12), and the discharg 
will be given by 


k.K..bH? = 3:087kbH? = K'bH:? cub. ft. sec. 


Thus corresponding to the values *82 and ‘87 of k, the values 
of K’ become 2°53 and 2°69. 

The validity of this formula receives remarkable con- 
firmation from the results of a large series of tests on such 
weirs, carried out in 1903 at Cornell University for the 
U.S. Geological Survey *. From a summary of these tests 
it appears that on broad weirs, for depths between 1 and 
5 feet, the coefficient K’ is sensibly uniform, increasing 
slightly with H. With weirs from 5 to 16 feet wide K’ lies 
between 2°62 and 2°64, while with heads between °5 foot and 
1:0 foot, K! varies from 2:73 to 2°64. Experiments at 
Cornell University in 1899, on a weir having a crest 
6°56 feet wide, with a sharp upstream edge ft, show the 
following results :— 


Value ‘of H ...... 5 1:0 15 2:0 5:0 
Rr AEOE Ke 249 | 9:59 | 2:53 | 2-47 | 2-69 


_ * Water Supply and Irrigation Paper, No. 200. U.S. Geological 


Survey. 
+ Trans. American Soc. Civil Engineers, 1900. 


oe] 


VILL. Partial Pressures in Liquid Mixtures. 
By WitutaAM Kpwarp Srory*. 

ie summer Professor Rosanoff called my attention to 

an investigation of the partial pressures in certain 
binary mixtures that he was making by the app'ication of 
the Duhem-Margules equation to experiments carried on in 
the Chemical Laboratory of Clark University, and to analogous 
applications that had been made by others. The mathematical 
aspect of the problem interested me; I studied it carefully 
and found that it was possible not only to improve the method, 
from a mathematical point of view, by the use of more 
convergent series than those heretofore employed, but also to 
extend it to mixtures of any number of components. Inci- 
dentally it appeared that the coefficients of the new series are 
more readily calculated from actual observations than those 
of the former series, that Raoult’s law holds for any number 
of components, that this law is an immediate consequence of 
the Duhem-Margules equation, and that Margules’ formule 
for the partial pressures in a binary mixture involve no as- 
sumption other than those involved in the equation just 
mentioned. The present paper describes my method in 
general, and its application to binary and ternary mixtures 
in particular. 


CoNTENTS. Page 
I Physico-chemical assumptions .......005.0600. 000s 97 
2. Mathematical formulation of the assumptions ...... 98 
3. The Duhem-Margules equation .............00ee ee 102 
eB OPEL AITO ere, ison 9) a0G si sie noi che Maik Houses Ma.oee suaveye hier 106 
5. General solution of the Duhem-Margules equation.... 107 
6. Special method for binary mixtures ................ 118 
ee De Maat ITM OUTS, bays ts. 5,4) ah! als Beavaletel. aka Rel E NN milked! oh 116 


1. This investigation is restricted exclusively to such a 
liquid mixture and its variations as satisfy the following 
conditions :— | 

a. All variations of the mixture shall take place iso- 

thermally, that is at a constant temperature. 

6. Hach component shall have a vapour, and therefore a 

pressure of its own. 

_c. The partial pressure of any component of the mixture at 
the temperature in question shall depend solely on 
the composition of the mixture—that is on the molar 
proportions of the several components,—and shall 
vary continuously when the composition varies con 
tinuously, being finite for all compositions. 

* Communicated by the Author. 

Phil. Mag. 8. 6. Vol. 20, No, 115, July 1910. H 


98 Prof. W. E. Story on Partial 


d. The partial pressure of any component shall be 0 if the 
proportion of that component is 0—that is, if the 
component in question is absent from the mixture, 
and only then. 

e. The rate at which the partial pressure of any component 
changes as the proportion of that or any other com- 
ponent changes continuously shall be finite; and the 
rate at which the partial pressure of any component 
increases from 0 as that component is added gradually 
to any mixture of the other components alone shall 
not be 0. 


f. Any component by itself shall have a perfectly definite 
pressure at the temperature in question. 

g. A possible association or dissociation of the molecules 
of any component shall be regarded as producing a 
corresponding change in the molar proportion of 
that component. 


It is important to observe the distinction between the 
partial pressure of a component of the mixture (to which 
conditions c,d, and e refer) and the pressure of the same 
substance in liquid form at the same temperature when 
existing by itself (see condition 7). In the latter case the 
component by itself may be regarded as the whole mixture ; 
its pressure is then the total pressure of the mixture. Other- 
wise the total pressure of the mixture is the sum of the partial 
pressures of its components ; it is in this sense that the 
partial pressure of an absent component is 0 (as in con- 
dition d), because such a component contributes nothing to 
the total pressure of the mixture. ‘These two cases—that in 
which the component in question is abs-nt and that in which 
it constitutes the whole mixture—are the two extremes, as 
far as that component is concerned. 


2. The molar proportion of any component of the mixture 
is the ratio of the number of “mols” of that component 
present to the total number of mols of all the components 
present. Let « be the number of components under con- 
sideration (not necessarily all present in the mixture), 
M4, Ng, Ng. ++ the numbers of mols of the several com- 
ponents in the mixture, 2, #2, v3,... 2, their molar propor- 
tions, and py, Po, P3,.-- p« their partial pressures, respectively. 
Let 


Ne 71+ ne ig 4 Benes) 


Pressures in Liquid Mixtures. 99 
then 
Np ae < 
Uy =— N? Ny = Nar, (7 = Ls 7 oe oe K), e (2) 
oe 
and 
Rttgtegzt+...t4, = 1. ° ° e e (3) 


Furthermore, if dz,, dx, div3,.., da, are the infinitesimal 
changes in the 2’s that correspond to any infinitesimal 
variation of composition of the mixture, we have, by (3) 


dads Par +... ad, = Or. 24. (4) 


On account of (3) the z’s are not all independent, as is also 
implied by (4), but when the values of any «—1 of the w’s, 
positive or 0 and having a sum not greater than 1, are given, 
the value of the remaining x will be determined by (3). In 
particular, if any x has the value 1, all the other w’s have the 
common value 0, and, if all but one of the 2’s are 0, that one 
isl. If x.=0, we have, by (2), n,=0 and vice versa, and the 
rth component is absent. If 2,=1, the rth component is the 
only one present, and constitutes the whole mixture ; then 
every n is 0 excepting n», When we speak hereafter of “all 
values of the 2’s,’ we shall mean all sets of values that 
satisfy (3). 

By condition c, each of the partial pressures p is a con- 
tinuous singly-valued function of the 2’s, finite for all values 
of the 2’s. In consequence of (3) it may be expressed as a 
function of any «—1 of the w’s, and, on replacement of the 
x’s by their values from (2), it may be expressed as a function 
of the n’s. It will be convenient to represent p,, the partial 
pressure of the sth component, by p<” when expressed as a 
function of the n’s, by po when expressed as a function of 


all the z’s by substitution of the n’s in the terms of the w’s 
from (2) in p™ (N fails out and the result of the substitution 
is just the same as if each n in p were replaced by the cor- 
eeeorgng z, because p® is a homogeneous function of the 
n’s of degree 0), and by p™ when expressed as a function of 
Bie B55 ey CE a ag : eros x,). It is to be observed 
that oe p® ) and oe ) are perfectly definite expressions ; but 


ps can be Peedi in many ways as a function of all the z’s, 


on account of the relation (3). Then, because p” is a homo- 
geneous function of the n’s of degree 0, | 
Op” op” OP Op? 
+ ng ot OL same 8) OS oe GD 
. On, Ons ie on ih Ol ) 
H2 


100 Prof. W. E. Story on Partial 
Also, by (2), 
PAC) (2) 
age yo. 
Ox, On 
and, by (3), 

»(*) (zt) .y,(#) (7) (2) 
Op, ORT eee, =n (SPs OP, 
Ewe oA prin On 

From (7), (2), (3), and (5) follows 


(r= 1, 2,3... 0, rn 


» (r=, 2, 3,-5.5 Senae : (7) 


oe 1 oxo + 2 ae + as oe tae 
es ea ee 
(2, + ty + 3+ +2...) 2] 
w= mae 4 nO 4 nO mPa 
oe (8) 


Furthermore, considering p‘”’ as derived from p© by sub- 
stituting the values of 2, 2, #3,..., @,_, in terms of the »’s 


from (2), we have, by (8), 


Op” Op” ya OP. DE) \ 
y= alee —(« Pen O22 ie ee a 
Op” { 9 
4 eae I (9) 
a Ly a ap... | 
N Oz, NA ; ) 


as also by ( ros 
By condition d, we have 


pe= 0 fore =0 orn =0 (s=1,2,3,... «). (10) 


Pressures in Liquid Mixtures. 101 


By condition e, 


(n) 
= is finite for alk-values of the n’s_ (7,s = 1, 2, 3,...,«) (11) 
7 


and Op 


On, 
From condition d or (10) and (9) follows 


nou torn =O... . ¢ (12) 


(n) 
OP a == TOr De 0 ‘Ge s=1, 2,3,...,« and r Ss), 


== 0 for #,= 0 Ges 1, 20, h— 2 and 73), (13) 


(n) 
Pe NE for = 0 (7s Ly 2y Soins 6 = 1): 


so that 


3° 
—————$——— > 


=—— is neither 0 nor infinite forw,=0 (r= 1, 2,3,...,«—1). 


Oz. 
From (9) follows 


Pa) pe i=. 2. OPE 1 Op”? 
ff IN © Oe, — N Oz, 


so that, by (11), 


for z,=0 (s=1,2,3,...,«—-1), (14) 


z) 
OP. is finite and not 0 fora, =O (s=1,2,3,...,«—1). . (15) 


From (7) and (11) follows that 


Ps) pe 
02, 


Pe Os bis 16 
T2555 0 th) ube 16) 


*) 


is finite for all values of the ws (r 
s 


By condition f, we have 
Pa elon a == Ty (oy 2) 8 yesg eyo (10) 


where P,, for each value of s, is a perfectly definite constant. 
for the temperature in question. — 


102 Prof. W. E. Story on Partial 


3. For any liquid mixture that satisfies conditions a-g of 
paragraph 1, we have the generalized Duhem-Margules 
equation 


ndInpy+ngdlnpotnzdinp3+...+n, dln p, = 0, (18) 


where “In” denotes “natural logarithm,” that is logarithm 
to the base e = 2°718..., and the differentials refer to any 
infinitesimal changes in the molar proportions of the com- 
ponents and the corresponding infinitesimal changes in their 
partial pressures, at the given temperature. This equation 
was originally given * only for a binary mixture, but the 
method of proof is applicable to any number of components. 
This equation may or may not hold for other mixtures, but 
we regard it as proved only for mixtures that satisly con- 
ditions a—a. 
On account of (2), equation (18) may be written 


a dln py + @2d ln pot xv3dlnpst ... +a dinp, = 0, (19) 
or, by (3), 


a, din 4eaod Ine? + ayd In at SS Ee +a__,d\nt«=} +dlnp.=0, (20) 
Pr Pr Pe Pr p 
which is a convenient form to use when 4, 2, @3,..., @_1 
are taken for the independent variables. This equation, being 
a homogeneous linear differential equation in the « functions 
Pis P2) P3r++ +5 Pp, Of «—1 independent variables, is equivalent 
to a system of k—1 homogeneous linear partial differential 
equations in these functions and therefore suffices, with the 
conditions (10) and (17), to determine all the « functions 
when one of them is known. But we shall find it more 
convenient to derive them all from another function to be 
determined by experiment. 
Because p, = 0 for x, = 0, by (10), there exists a definite 


positive power of w,, say a*,—where e, is a positive integer 
or fraction,—such that p,:.,° is neither 0 nor infinite for 
x, = 0; sv that, by conditions c and d, this ratio is neither 0 


* Duhem, Amn. de l’ Ecole normale sup. (8) vol. iv. p. 9 (1887) ; “ Disso- 
lutions et Mésures, 3 mém., Les mélanges doubles,” Zrav. et Mém. de lu 
Faculté de Lille, iii.» (1894): Traité élémentaire de mécanique chimique, 
vol. iv. book 8, chap. 7 (1&99). 

Margules, Setzwngsber. der Wiener Akad. vol. civ. p. 1248 (1895). 

See also Ostwald, Allgemeine Chemie (2 Aufl.), p. 636 ff., and Nernst, 
Theoretixche Chemie (2 Aufl.), p, 118. 

A simple deduction of the equation by Luther is given in Ostwald’s 
work, p. 639. 


Pressures in Liquid Miztures. 103 


nor infinite for any values of the 2’s. Regarding the p’s as 
functions of all the z’s, put 


Ps 
In a =U, (s = Le 2, ey seey K) Reso ae (21) 


then u, is, by condition c, a continuous singly-valued function 


of the 2’s, finite for all values of the w’s. From (21) 
follows * 


ee ieee oe Se ul. |. (22) 
where, by (17), 
eaten eb (aa A But eM )< » (23) 


Now let «,, the p’s, and the u’s be expressed in terms of 
#1, 2, L3,.-., 4-1: Then, for 


a Os =F 2 Sys « =" &— L); 
we have, by definition of the derivative, (22), and (15), that 


op, |\ DP, e.—1 


| tee e's is neither 0 nor infinite: (24) 
v wv 
&§ s 


therefore, because uw, is finite, by (21), 
eae (9 ae he es. (2d) 
Also, for x, = 0, that is 
£,+2,+4%,+ ...+2#,_,=1 and n,=0, 
we have, by definition of the derivative, (13), (2), (22), (11), 


and (12), 
d cy we ee 
i =N 2 = Nfs = os = ,fx—lg"« is neither 0 nor infinite ;(26) 
av n « 


therefore, because Ux is finite...., by (21), 
eve Se eS een eee £1 


K 


* This is substantially Margules’ formula, p, = P, 2° eu, where u = 9 
for 2 = 1 (see loc. ctt.). There seems to be an idea in the minds of some 
that this formula involves an assumption ; but, clearly, (21) simply defines 
the use of the symbol w,. The only assumption that Margules makes in 
the formula as he gives it is that u can be developed according to positive 
integral powers of 1—z. 


104 Prof. W. E. Story on Partial 
By virtue of (25) and (27), (22) becomes 
ene (6 = 1, 2,3... sagas . . ae 
where, of course, (23) still holds. From (28) follows 
dp, =-(da,t+eadu,)je"s (3s = 1; 2,°3,..,,%) 20 tee 
and, therefore 
main, = dz. aduy ts =a, ee 


so that (19) becomes, when we take account of (4), 


2 du, + a du,+a2,du,+...tadu =0, . (80) 

or, by (3), 
x,d(u,—u,) +a,d(u,—u,)+ a,d(us—u,)+...+4,_,d(u,_,—u,) +du, = 0, 
ss (eee 


if we express everything in terms of 2,, 2, @)..-5%,_1 
On changing the signs of all terms of (31), distributing 
—du, equally among the other terms, and adding 


i. 


k—l1 


(du, +du,+du,+. Me +du,_,) 


to both members of the equation, we obtain 


1 1 1 
a _ 1) d(uy—u,) + (=, _ 2) d(ty—u,) + = — x;)d(13—u,) 


1 1 
+ et (ps) tes) = Pe a alate +h ane 


Putting 
1 1 
i forehead Lge | ~2 (s=1,2, 3,7..,0—l)aee 
and 


i! 
ay (tet ate FU) = . * £ (34) 


taking 21, 20) 23) ++ +5 %,- for new independent variables, in 


Pressures in Liqud Mixtures. 105 


terms of which the a’s, w’s, p’s, and the new function @ are 
to be expressed, we have *, from (32), 


2d (uy—u,) + 2d (ug — u,) + 23d (Uzs—¥, J +... +2, dU, — u,) edo. 


‘ 


As #(s=1, 2,3,...,«—1) may have any value from 0 
a 

gee + — so that 
the absolute value of z, is always less than 1, which is 
decidedly advantageous when we have to do with infinite 
series in the variables. Namely, the smaller the absolute 
values of the variables the more readily is the convergency 
of such a series determined and the fewer terms will it 
probably be necessary to use in calculating its value to any 
given degree of accuracy. Further on we have given a 
special method for treating a binary mixture, which is 
practically equivalent to that mentioned in the last footnote ; 
by means of which we have calculated the formule (26) 
from actual observations of several binary mixtures. We 
have also calculated these same formule in terms of the «’s, 
and the special method not only has the advantage of using 
variables with smaller absolute values, but gives series for 
the u’s with more rapidly diminishing coefficients than the 
former. It does not, however, seem possible to predict that 
the series for the u’s will always be more convergent, or have 
more rapidly diminishing coefficients, the smaller the absolute 
values of the variables involved. 


K 
to 1, z, may have any value from — : 


* The method we are going to apply to (35) to determine 
Uy) Us) Usy--., U, from w as functions of the z’s may be applied to (31) 
to determine w,, U,, Us,..., %,—1 from wu, as functions of the 2’s. Also, 
if, after changing the signs of all terms of (31), we had distributed only 


xk—1«-ths of —du, among the other terms, had added 2 (du,+du,+du,+ 
.. + +du,) to both members, and had put 

h of 
Ls=— —z,(s=1,2,3,...,x«—1) and 5 a Fly FUs+ ted Vi ay 


we should have had the same equation (85) with the new variables z 
and the new function . The determination of w,, w., us, ..., uv, and 
this function » in terms of the new 2’s will follow the same lines as in 
the text. For a binary mixture (« = 2) this method is preferable to that 
given above, because here the absolute value of any * never exceeds 3, 
while above it may amount to | (for the corresponding x = 0). But for 
values of k>3, the method of the text gives the smaller maximum abso- 
K 


: K-22. —1 J 
lute value of any z, namely ear instead of Re (for the corresponding 
ls 


ro) 


Or 


106 Prof. W. E. Story on Partial 


4, For #, =0@ =1, 2,3,...,«—1) we have; by Gay 
and (24), 


soa (I+e ort =e", 


Ox, wer 
Op. FOB, 
—=g¢—"e*=—0 (k=1, 2,3,...,e—-landker, 
Oet0 Lor, ( ? 
so that, by (16), 

Ou 

te = 0 (k= 1, 2. eel) 
an, ( ) 


and, therefore, 
of du = 0'5 en i (ot ee (36) 


also, for 2, = 0 we have, by (29), (4), and (26), 


so Beni( —1+e reset 6, hie? 5 
SO ive by (16), 

0 (k (Ly Dy Sue sige oD 

Ox, 


and, therefore, 


edu =O. 26 9! 2) 3) 


It is not self-evident that 2 du, =0 for w, = 0, because 
one or more of the derivatives of wu, might be infinite, but 
(36) and (37) here proved show that even if = is infinite 

I 
for z, = 0, it is infinite of so low an order that its product 
by x, is 0. 

If, now, 7, =1(s=1, 2, 3,....,«) all the other 7 ae 
0, by (8), and, therefore, by (36) and (37), equation (30) 
reduces to 


du.= 9.5. » + «se 


Pressures in Liquid Mixtures. 107 
Therefore, by (27) and (23), 
pV aeons — lh (= 172.3, .-.,«); (39) 
that is, by (4), 


ace Ae nD, Kt a re s 


\ 
my 

ls (40) 
) 


a ae Peer, (6b 2. 3,3 1). 


Equations (40) express Raoult’s law, which is thus seen to 
hold for a mixture of any number of components and to be 
independent of any assumptions excepting those made in 
conditions a—g *. 


5. Going back to equation (35), we assume that the func- 
tions wu can be developed according to positive integral powers 
of the z’s ; this is equivalent, by (33), to the assumption that 
the w’s can be developed according to positive integral 
powers of the a’s. Then, by (34), @ can be similarly 
developed. Any term of such a development is of the form 


Ps Maer Tp: A 
aa “9 <3 See. EST 


multiplied by a constant coefficient, where each of the 
exponents 41, Jo, Js.+++) Je-1 18 any positive integer or 0. 
Let the coefficient of the product of powers of the <’s just 


TI in u. be denoted by a” and the coefficient 
aE Ue a ae nee ae 


* Gahl (Zettschr. fiir physikalische Chemie, vol. xxxiii. pp. 192-195) 
has considered what might happen if the last part of our condition e were 
not satisfied,—but his cases are purely hypothetical. Considering only a 
binary mixture, he assumes that the partial pressure p of one component 
is proportional to a power of the corresponding molar fraction « whose 
exponent is an integer as great as 2,whereas it is not certain even that 
this pressure can be developed according to integral powers of 2. He 


says that it often happens that oF = 0 for « = 0, but cites no specific 
case. It may be well doubted whether a mixture can have any com- 
ponent for which this condition is satisfied unless p = 0 for every value 
of x, in which case the component has no pressure of its own and the 
Duhem-Margules equation is not proved for any mixture that contains it. 


108 Prof. W. E. Story on Partial 


of the same product of powers of the z’s in w be denoted 
= 7 
by 91> Ja Jar ++ 5G, 4" Then 


ape (7) Gi oF 5 os Gn-1 
ur = ay G91, Jor 930+ ++ In 11 a“? Rees et ) (41) 
and 
— Ah 12 ~93 Sei : 9] 
alia 2» €911 Jay Jas Geni=1 72 =Z se n-1? (42) 


where %, denotes the sum of the terms following it 
for all possible combinations of all possible numbers 
Yiy Y2) 939+++59,—-, Of which each is a positive integer or U. 
For the sake of brevity, we shall write @® instead 
(7) ‘ 
of a : 
eee a and co instead of Co a, on A 
Any coefficient with a g written among the suffixed dots 
with +1 or —1 attached shall denote the result of increasing 
or diminishing that particular suffix by 1, without altering 


the other suffixes. Thus, a”, _, denotes the coefficient of 


) 
I.- 


u. that is derived from a” 
4 D192 Gor Jar+ ++ Ie 


instead of g, for the k-th suffix, and ¢..y,41.. denotes the 


by writing g,—1 


coefficient of w that is derived from ¢9,, 95, Ja.++s%¢-1 bY 
writing g,+1 instead of g, for the k-th suffix. I 


Equation (35) is equivalent to the set of «k—1 homo- 
geneous linear partial differential equations 


“1 ao BO gO) 4 ae, Oe 


| 
in Oér ey Tee | (43) 


Substituting the expressions (41) and (42) for the w’s and 
w» and equating the coefticients of like products of powers of 
the <’s in the two members of each of the equations (43), we 
have, for each set of values of 44, Jo, 93, +++ 9,1 


k=«-1 (k) (x) \ 
Sg: a,. pts yah pena SS ye = pCosed | 
2 (409,71 y-1) = 9 +. id) 
(7 = A, 2, Bye..,K—1), y 
k=x—1 


where 2 denotes the sum of the terms following it for 


k=. ( 
all values of k from 1 to«—1. Namely, the term actually 


Pressures in Liquid Mixtures. 109 
written in the left member of (44) is the coefficient of 


aft 283 ee ea ae 1 

haat ts Se Voy hes 
in the k-th term of the left number of the r-th equation (43) 
and the right member of (44) i is the coefficient of the same 
product of powers of the z’s in the right member of the 
y-th equation (43). But it is to be observed that the 
numerical multiplier of the 7-th term of the 7-th equation 
(44) (the term for which & = r) is g,—1 and not g,, and that 
the k-th term of the left member of the 7-th equation (44) 
is to be omitted if the k-th g of the set in question (that 
is g,)is 0. The r-th equation (44) falls out if g, = 0 for 
the set in question. 

The left members of equations (43) have no constant 

terms and, therefore, the derivatives of w with respect to 
Z1, &2 23, -++> Z,_, have no constant terms; that is 


Cum = OF (k= 1, 2,3,.--,4-1). - . (45) 
where ¢,, denotes the coefficient of 2, in » (the ¢ whose 
suffixes are all 0 except the k-th and that is 1). Further- 


more, if a’? and ¢, are used for brevity to denote the 
constant none of wu, and w, respectively, we have, by (34), 


OG, ade te a pa fee, (i . (46) 
If we write, for any given set of values of 9, Jo, 93s ++ +5 I-45 


NtGPtGt t+ KRaiHG& - «+ (7) 


and add the system of v equations (44) for this set, remem- 
bering what we said about the numerical coefficient of the 
term of each equation that corresponds to the number of that 
equation in the ou we find 


k=k 
(G—1) > oo TC ee . Sci G Crees (48) 


On ee G—1 times the 7-th equation (44) from 
as times equation (48) we have 


wi God) ie the 
(G Lie ae Gat) i ae ee 


or, writing g,+1 instead of g,, preserving the notation (47), 


G (ao? nag) ) —_ (9,.<+ 1) Cig +1.- : (7 = by Z. Deas ,«—1) * (49) 


sha a @) sy ) ere ee 


110 Prof. W. HE. Story on Partial 


for any set of values of 9, 92, 93,--+59,_, that are not all 0. 
Namely, it is evident that the constant terms of the u’s will 
not occur in equations (43), being driven out by differentia- 
tion; they will be determined later. It may be remarked 
that the notation (47) enables us to write equations (45) 
thus : 

c =O if Ga Ce ee 


, Pe 
[Wee poet ule kaa Cag ti. (= 1, 2,3,...,e—-LandleG), (51) 


k=x—1 
i 
fee =) aa2 (x) eee : 
e—De (e—1)ae Gos Fe APE he daly 
from which follows 
(x) 1 k=xc—1 ea 
gi ae lt (eG 2 Gt eg41.. (92) 


for any set of values of 91, 92, 93,-++59,_; that are not all 0. 


Substituting a“? from (52) in (51) we have 


er eee 


1 k=x—1 g +1] \ 

0 1 re Aaa | 
_! Saha Fase: («—1)G = OTe ae a3 G gia t (53) 

(or ls Bis eee eoals ) 


for any set of values of 9, 92, 93,--+59,-, that are not 
all 0. It will be observed that the aggregate multiplier of 
Oa in (53) is 
OAL. gl aie ae 
(x—1)G he ogee dG 


(54) 


The coefficients of the w’s are all given in terms of the 
coefficients of w by (52) and (53), excepting the constant 


terms. 
We have, by (23), u,= InP, for 7 = 1, that is, byxooy 
K—2 1 
for, 2.= = joy and bs ee 1, 2; 3... 


Pressures in Liquid Mixtures. 111 
and s=r); but u.=In P, for 7, =1, that is, by (3), 
for vy = v3 = v3 = ... = WW | 


orfor 2, = SG =, 2 ayais 1). 


? GL 


Substituting these values of the 2’s in (41), we have 


that is, if 2 denotes the same summation as ¥, with the 
126 


omission of the term for which G=0, 


r Ir (K—2 
a? =In P,— 3 (—1)" eNO, (1% B+. -1) OS 
1=G 
and 
a’ in Po sete 2a 5G 
me ae ee (56) 
Substituting the expressions tampa. 5) 2G) from (53) 


in (55), using + 2 to denote the same summation as > with 


the omission of ie terms for which G=0 or 1, and taking 
account of (50), we find 


Oh P, + 2 g “Dee iailaw = ie 
sin Bae eye sale G-1  («-2) (G-1) 
(e—~1) 9, 
(ea G—D 
es 


=In P+ — 1) ——__, + — (P= 12 By tg ELS (aT 
2M Gayse=n" ae 


Substituting the expressions for qa” (l= G) from (52) in 


(56), taking account of ae we find 


(x) __ G 
ay =InP.+ > Soldewaiee 
: 2=¢ aie 5 ee [ ao Gal 
| 1 
=In Ph esis (Gan (Gayo (58) 


112 Prof. W. E. Story on Partial 
From (41) and (51) follows 


oe Fy We 2): Oe eae 
: 2 


By ULce oeere Ir > 
uu =a a+ ps Cun Gye Thon Oa) ae Se oe (59) 


0 
1=G G 


(r=1, 2, 3,..., «—1), 


where, by (57) and (58), the constant term of the right 
member is 


eis P, Bo fe ee 
ay —ay =In 5 + > cae ae Cue - (60) 
)) ast 


Also, by (28) and (33), 


uy—U,= In eae Sit cs: Kaine (61) 


Pr&r 


Equations (59) and (61) serve to determine w and ulti- 
mately U,. re 1 2, oy oe 84 K—1) and the formule (28) for 
the partial pressures in mixtures of any given components 
from actual observations of mixtures of those components in 
different proportions. Such an observation is supposed to 
give the values of 2,, y, %y,- ++) Uy Py? Pes Po * Pur Pat Pry 

-> Pe-) : Pe From these values are determined 
—u(r==1,.2,3,...,e—1) by (61) and 2, (k=), 2/5 ee 
x—1) by (33). On substituting the value of u.—u, for any 


value of 7 and the values of the z’s in (59) we obtain a linear 
equation in the coefficients ¢,..... for 2=G and in the con- 


stant Pa —a; each observation gives «—1 such equations 


in the ¢’s and in the «—1 constants, corresponding to the 
«—1 values of yr. So far as we know, is an infinite series 
in the z’s, but in practice we must suppose that it is con- 
vergent and that we shall get a sufficiently close approxi- 
mation to it by taking a certain number of terms of it. 
As there will generally be no reason to assume that there 
is any difference in the order of magnitude of the different 
z’s and, therefore, of the different z’s, the terms of w that are 
of one magnitude are those that are of one degree in the 2’s, 
that is those for which G, in the notation of (47), has one 
value. ‘The natural mode of procedure will, then, be to take 
the aggregate of terws of w whose degree does not exceed 
a certain number as a sufficiently close approximation to the 
whole expression. In other words, we agree to neglect every 


Pressures in Liquid Mixtures. 113 


coefficient ¢,..... the sum of whose suffixes exceeds « certain 
number. What the limit of the degree of terms considered 
shall be will depend on the accuracy and number of the 
observations. The number of observations must be at least 
sufficient to furnish as many equations (59) («—1 for each 
observation) as there are coefficients of w to be determined 
plus «—1 (for the «—1 constants a —a\), It is preferable, 
however, to use a much larger number of observations and to 
solve the equations (59) by the method of least squares. 
The larger the number of observations the more accurate the 
values of the coefficients calculated from them may be 
expected to be. In fact, by increasing the number of obser- 
vations, formule can be obtained from which the partial 
pressures for given molar proportions of the components can 
be calculated much more accurately than they can be observed. 
But, to effect this increase of accuracy, it will be necessary 
to carry out the numerical computations to several places 
more than are given by the observations. 

When the c’s for 22G have been determined from (59), 
the as for 1=G can be calculated from (52) and (53), 
with due regard to (50), and then, by (28) and (41), 
ay” (r=1,2, 8,..., «) can be found from any one observation 


Pr (7) g g g UO 
in —_— j— > a hea lish we well Sa Geo 2, Dae aig Wop (62) 


Finally, P, (r=1, 2, 3,..., «—1) and P, are found from 
(57) and (58). 

After the values of the a’s have been calculated, the 
partial pressures for any composition of the mixture can be 
easily found from (62) without going back to (28). 

The constant term of » plays no part in the determination, 
but, if wanted, its value, by (46) and (57), is 


1 
a area In (P, Ie Vas te Pet) 


Cora 1 Fk(p—9\7e 6 
5 = G@—DSt (G1) 2 ( 1) (x 2) 2 ( 3) 


2=G4 


Special Method for Binary Mixtures. 


6. For a binary mixture, the alternative method given in 
the footnote to (35) is preferable to the general method 
developed above, but we give still another method for this 


cal 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. 1 


114 Prof. W. E. Story on Partial 


ease. So far as they correspond, we designate the forraule 
by the same numbers as in the general method, with an 
accent attached. 


By (28) and (23) we have 


p=, e", Po=ag ee, + + (29') 
where 
win P\ for #j=1, u.=|ln P, for ej=1)) ae 
1 1 ry 
Put = 57h 2, Coes (337) 
and 
Uy —Ug=O. go... 5 
Then 
ste)duy+ (5 —2)duy=0 30) 
(st2)dn+(, eo ys ee € ) 
so that 
1 1 ; 
d= (5 -=) dla, duy=—(5 +2) de | ake 
Put | 


m=, a” a, Uy 2 a®) a, 


oxnase A ee »)/ 
w= Cyc! . (42) 
Then, by (64), 
Spa POLLS ge, zye, ce, 
l=y 2i=, l~<y 
x. 
=, .@).9~1 agi mTr 
ay Iie =— 92 oer —2ICe, 
l~=g l~=<g l=y 
so that 
1 g—1 ) ! 
1 
a Woke = g “gare Cg-19 . . (So ) 
ot Wiles oa 9/ 
a= 2°9 g-l) " . (52’) 
aud, in particular, 
] 
(1) (2 “aie 
os 7 1 Ge a= see 


Aliso, by (23!) and (41'), because <= 7 corresponds to xv, =1 


a 
: 


Pressures in Liquid Mixtures. $15 
1 
m2 — 3 to fab, by (33), 
ye 
by Weby BOY AC ya Ont NOY 
=n Pi- i a [nn By = 2 yt sae (g+1)27 
accep eee (57") 
= n eS . « ; 
"yy 2 (9g +1) 
a = 2 lou g ge, 
=In P,~ ¥(-1%a ae ae a =, bl ee 
at ) l=g oo 1=g (y+ ne 
Stee S(t) 58! 
ot = oo cet 1) oi 
Now, by (42'), (34'), and (28), 
oI = =) 61’ 
>,¢,7=ln & yt (61) 


by which any desired number of the coefficients ¢, can be 
calculated from a sufficient number of sets of corresponding 
values of 2, 2, 2, pi, and p 2 From these values of 
the c¢’s the values of oe and Oe for g as great as 1, and 
not greater than the greatest sce of any c by more than 1, 


can be calculated by (53!) and (52’). Then, from any one 
set of corresponding values of 2, p,, and pz, we have, by 


(28) and (41’), 


ah? == inet — > a et, aint —2> Goes (62') 
Ly l=q v3 l=g 

where, by (53') , (52’), and (61/), 
> Peta Boe! - y+ e, #4 =(5— z) Be, #42 
Le g* eae g 2 l=g 2 A? 
=| In (2 anh ] +2 ey 

L Po 7) : cen rale 

] g af ) c 

Ba 2 — = > «, 7 = A= —(5t+71> «#42 Zi 
l=g - 2 =, ° Zoi’ 2 / \=9 ; t 251 

a 2) aM ‘| , 

=—2,|] > 3 

1 E ae +23 


116 Prof. W. E. Story on Partial 


so that 


iar =, In ee +a,[ In (22) +00| —:> Pari } 


1=¢ 


P tg . Be 
a ey [In (2) =« | + Wy in) ae = G+ 29, 


from which follows, by (33) and (34’), 
a + q=2 E 1n(2) + a in(”*) ] —~ 23,04 he (66) 


a —a = cp. a et eae ET eel a) Rae ee oe (67) 


1 2) e e . 
Then a ) and a are determined from a single observation by 


(65), or by (66) and (67). Finally, P; and P, are deter- 
mined by (57) and (58'). 


Ternary Mixtures. 

7. For the sake of showing how the general method 
works out in practice, we give the formule and equations 
for the case of a ternary mixture («=3), neglecting terms 
of w that are of higher than the fourth degree in the 2’s. 
This implies, by (52) and (53), that terms of 1, up, us that 
are of higher degree than the fourth in the z’s are also to be 
neglected. The formule shall be numbered as in the general 
method, with two accents attached to each number to distin- 
guish this particular case. 


a — yeu: —y es. " 
P20", Po— %2¢ 2, P36 5 ° . . e . ° e ° (28 ) 


u,—InP, for «,=1, u,=InP, for #,=1, U,—InP3 for v,=1; (23") 


1 1 39" 
— —~—Z A i es a) €,=2, +%,5 . . e e . . ° ( ) 


ast ies aaa a 

1 Q4/I 

g(utu)=o; Si Pgh) 2 Saat Senate a a Senin (34 ) 
1 Leia! ey (1),2.1 ,Y234 qU)z2 

t= od + az, Ae de® at ay Zo 11 212 Mq9%9 F M3971 +4 2122 | 


1).4 
1 ).3 4 (Dot 4 q)z8 (1) 92 1 ge 234 qUzt- 
az 2+ a 28+ aget + tay M29 t Opp 779 F371 gt U4 “9% 


2 2 2), (2)3 (2) 2 
1 eed a” + az a ae a a a4 + azz, + an a +a 50 a + as Z3%2 


2 74 (2 3 4 - 
ae, + a+ ales + asede, + ayy ?25 + 29 %1%9 + os 2 


3 3 3 (3),2 4 g8)z34 q@) 22 
U3— ay aso% a ane 2 e as? i ¥ aye 1*2 +4975 T3921 7 Soir 
3) 4 , 


(3), 3)z3 4 q(8)04 1 92) 232 (3)9.2 1 g@y 3 2 
+a ae 224 aes t atdet + agate, + ay Met 37172 TMM 23 | 


~ Pressures in Liquid Mixtures. 117 


a if 2 
W=Cy+ 6, 9% HO ZA la ty F422 FH Cy g25 + Cg 2? + 6272, ; 
2"’) 
2 3 4 3 2B (42 
HC, 92,25 + Cyg2o + 64921 + 0512525 + Cy.2 125+ 652,25 + Coo 


(Messe or ei a ee rr me (50'') 
te eG aCe RRND! wale oa a wp) (AT) 


a) 1 sayilill: oD Hee } 
Pig “29D 11 Cap “1a eon 20 4 Cs9— 4 “a1 | 
a) = 1 ah 1 3 a Me 1 
aang! ae 9°21 51” yy = 02+ qa A Cogs 5 +3 ~ g au 
(1) 1 AU ee al iL (1) + 2 | 
Ay, = Cy) i 9%31— 3% %p2 =Cipt 3 3022 9 Cyg2 Ugg =Cy3 + 6 713 — 3 “as : 


(1) (1) (ay all (1) a eu 


Mag =C497 Ug) = C31, gg = Cn99 Ag —=Cj39 Mg = 45 


= ol 
1 By ona cy 
(9. ya) ; (2) 
eo r 5 ou Gg — Teg eu 1 G2 3) = C29 — 4 eso F qo | 
| 
) [ee (2) gO 
<j baie Hens 5197 “2 =—%2— | seat 3 Si 3) 30 8 tot g » C319 | 
nae 1 1 1 | 
a ee Ss (2c as a qa — 
Sly — 5x1 T 3 Cy99 Ay Cg — BKaat 9% 39 3 = o3— GF sont 5 3 049 | 
(2) Quan. Oe Gy es 
— My Hyg, 3) = F319 Ang = Cn99 yg = 0132 Ag = C04 > d 
yi. i! ee = a \ 
15 foot 5 = e119 icf ah Coos Oop = 20 Z Cay — o> 
{ 
(3) 1 1 @y2- a! (3) _ 2 | 
ae iy 5 Ca 5 Pi Poa lean ain yk ial OMe ONL BD. 3 Cay 2319 + (52!) 
BO. 1 BE. ym 1 1 (8) 1 1 2 ¢ | 
21 — °21 oe ae 3 7297 A)5 =%2- BGT: 3 13° B53 —o3 — 618 304" | 
me, (3) (= (a) eo (Cy eae (Gy, aee h 
Be 49 — "402 3, — 31>, 99 Cnr = yg — F132 = gg — "045 ) 


Q 1 i! 1 
4 =In P,+ 4 (Cop — 041 + 99) — H (C39 — Cy + C9 — Cog) + 48 (C49 — C3) + C99 C3 + Cos)s (57") 


.——--+-- 


1 1 
» = In P+ 4 (Cog — C11 + Coa) + 76 ice Cy + j9—C93) + Ze Ze (C49 —€31 F 29 C13 + Cos)» 


1 1 1 
fin P Boke 4 (Cog + Oy) +92) + 16 (C39 ++ Cy) + Oy + ss an 48 (C49 +65) + 622+ 13+ %4)3 (58) 


118 Prot. W. E. Story on Partial 


putting 


@_ g@— 39, aa = 4 : 
eal == 0. ° a. = bys) os 5 
\ 
by +2 stake rat St alge On? 
QT 2eyyz1 + GiZ2F 5 30% Fate 5 M1227 23 M4071 T 1 125 
| 2 9 1 2 Py %, 
a aD a ! 
ot = Coo% 1% F Bite er In p,@ \, | (59"") 
fet 
| 1 o 2 1 3 2 9 
2 : ae oe Pe eee Le mag 
Ay Heyy 2 A 2Uyozo +5 Caz Feet 5032 3 C41 tT 3 C2971*2 
Pas ; 
A / 
TF 63%) % 245 50422 = In (Con (61) 
P3"3] @ 


- ] ] 1 
C) ea aes ; ‘ : 
6,7 = In ( =) ee aie 3 (“so S12) — 54 (a1 + M13)s 


r 
C2) ale 2 
by a in G 


1 1 1 
) fea haa @ (Ca +03) — 4 (a1 +13) ) 


) 
0"). 
) 


c= EY n(P,P,)-+- 4 (ip 61, P09) te eu 631 + l99— Cig Cpa) 3 (63") 


a 


) i: oy 
u, = In ( *) «=In ) u, = In . (21") 
Vv, - Vo tg 


Equations (59) and (61) are here combined into one set 
with the double number (59!) (61). Hach observation 
furnishes a pair of such equations. Together they involve 
14 constants to be determined, namely, 


q) (2) 
55 “» Coq» a0» C4» 29's Cons Cog» oar Cray Cor» C199 ©3192 Co29 Sia 


and seven observations will, therefore, suffice to determine 
these constants. But it is better to use a much larger 
number of observations and to solve these equations “by 
the method of least squares. This solution is effected as 
follows. 

For any given values of g; and g, put 


Pah oD Be 2M Pivs oh ~92 Pay 
24 oy = [ng], Zale 1 a(S *) = [99h 22 Zod (= a = [1 Iel- 


where 2 denotes the sum of the expression that follows 
ODE. 


[70 face page 119. 


(a) jb) 
bs s Coos Cyos C4o° 0 e Coo: Cos Coa: 


n 10] [20] [30] 
210] 4[20]  3[380] [40] 
[20] 3[30] (40) 2150] 
[30] ${40] 250] +160] 
n 101] 502] $08] 
2[01]  4[02] 303] S04] 
S[o2] 3[03] [04] —2[05] 
[03] 5[04] 2[05] [06] 
[ot]. fa1]» Fel] 81] te] at] 12] ee 
[11] 221] 5131] 41) 520] (21) 322) 323) 
soo] [12] [22] 2fae] = [1] fla] 18] 4 
[21] 281] S[41]) $[51) 3[30] [31] 582] 283] 
12] 3122] (82) [42] [21] 322] «= (23) S24] 


503] 313] 523) [33] [12] fa] saa] 8 


Pressures in Liquid Mixtures. 119 


it for all the observations. In particular, if n is the number 
of observations, 


[00] =n, [9,0] = a x. 0g.) = 22 or 


(00],= =n(2™), [9,0], = = <n (28), eGo) 
ave obs. 


Se) 
00},= Zh (ns >) = 2 4'n (2) =D etn ( B 
[00),= nl ty)” [7,91, as " P3ko) Lo; obs. 2 i } 


In addition to [00]=n there are 47 of these sums to be 
calculated from the observations, namely, 27 sums [4g g2| for 
all values of g,; and gy whose sum does not exceed 6, 10 sums 
[71 ge}, and 10 sums [94> ]9 for all values of g, and gy whose 
sum does exceed 3,—as is indicated in the adjoined table of 
normal equations. 

Each line of this table corresponds to one of the 14 normal 
equations ; each of the first 14 columns contains the multipliers 
of the constant that stands at the top of that column in the 
14 equations, and the last column (after the sign of equality) 
contains the right members of the several equations. The 
first 27 sums mentioned above occur only in the left members, 
and the 20 other sums only in the right members. ‘The 
multiplier of any one of the last 6 Coney ints in any one of 
the last 6 equations is the sum of two terms, as is also the 

right member of each of the last 6 equations. It will be 
; noticed that the first four constants do not occur in the second 
| four equations, nor the second four constants in the first four 
equations. The simplest mode of solving these equations is, 
therefore, to eliminate the first four constants successively 
from the first four and the last six equations, and then the 
second four constants successively from the second four 
equations and the six equations resulting from the previous 
elimination. When the last six constants have been found 
from the six resulting equations, their substitution in the 
equations previously obtained by elimination from the first 
four and the second. four equations will give the first fonr 
and the second four constants. The advantage of this order 
of elimination is that we never have to deal with more than 
ten equations at once, and that we do not have to introduce 
any constant into an equation in which it did not previously 
occur in order to eliminate another constant. 

When the c’s have been thus found, the a’s are easily cal- 
culated by (53") and (52"), excepting a, ae. and a, which 
are found from (41) and (21’) by means of the other a’s 


[To face page 119. Pum, Maa., Ser. 6, Vol. 20.] 
NORMAL EQUATIONS. 


FOU eee a (Or, eC ommmcr Cae Gh C5 Cop: Cia 
mn 2[10] S20) 4130) a2 : (ol) [11] 3102] : [21] all] 1103] = 00), 
210] 420] 3[30] $40] [1] 2[21) [12] 2[81] [22] 2113) = 10], 
S90} sso] $40) 2150] 2) 331] 22] 41] [32] 193) = Spry, 
430) [40] 2150] G12] ae!) atu 32 sft ioe (ea) =f 

n 201] 5(02] 3[03] 10) 3120) [11] 5[30] (211 [12] = [00], 

sor) 4f02] 303) $[o4], 21H] (21) 2[12] 331 (22) 2[13] = 01], 

3f02] 3|08] Sfo4] —2[05] 712] 322) $[13] 3[32 [23] 514] = 3(02], 

fos) Sox] 2105] foe] 303) +{23] 14] 4(33] 524] 415] = 403], 
fo) 21) S21) 431) [lo] >U11] [12] a13] (i e2l+ [2°] [12]+ 330] s03)+ (21) [22] + 5[40] 313] + $[31] qosl+ 22] = [01], + [10], 


ma) 2m 43 4) Fe eH 203} | B}+ 330) [22)+ ;{40] is|+4au)  (B2]+[50) 3128)+ 541) 44}+ 9132] = (11), +5[20] 
Moz] [12] 3122] 2ae] [11] 112) 313] S24) (081+ [21] 3[13]+ 31] jfo4]+ [22] [23] + 541] Ai14]+ s[32]  2f05]+ [23] = oz], + [11] 
pr] Bn feay fo 480) BL] 584) 3) | B+ 3(401 [32]-+ 250) Yesy+ Man) (H2]+5{601 FL88)+ GE ‘osj4 a4] = [21], +3[80] 

= q(12],+ 5121] 


2a) eq) 2) SPH ‘en a] Feay i+ GaN 328+ Maye 232] 5831+ 501] 241+ gL? [15] +5133] 


os] 3s] 523) 388] 2] 28] sas) 505) m+ (221 Ura}+ 432] Y[05]+ [23] 42414 4421 2[15)+ [33] g{06+ [24] = 5[08],+ [12] 


| 
| 


| 


‘ 


120 On Partial Pressures in Liquid Mixtures. 


and the 2), 22, 1, P2, p3 that belong to any one observation. 
Finally, P,, P., Ps are determined ‘by (57') and (58"). 

Jf wanted, ¢, is calculated trom (6: cad 

The most laborious part of the calculation is the deter- 
mination of the 47 sums that enter into the normal equations. 
This is best effected according to the following scheme. 

For each observation, find 


t X3 ) 
Pabre dng, ine. Une,,) dni’, la; In vail In rd 
Vy 2 Ps Ps 
LX; Dod 
In (Pi) = hy, In (228) = les 
P3Xy P3®2 
next, by successive multiplications by z,, 
SE PREC Pe 3 asics 
~1 cae “9 Es e a 6 = Ke 9 
oy ge el, = 23 
‘ Beil tik) : 
al,, 2il,, ail, — [test i = 4xl,];5 


then, by successive multiplications by zo, 


ay Bas) Sas ees ee [test 29 = <5 x 23], 
Mikaa ikay Fakes Sha. Sake [ test 2,23 = 2, X 29], 
Pies Races Mikey eee [ test Pin Se, KEE ob 
Ray Rite, Meee [ test 2f2) = 2, x <f25 |, 
a re [ test Sabn = 2, XE 23 |y 
29 [ test ohz, = 2, X 242) |- 


[test 231, = 2)xl,], 


Reads) Ryeel, [ test Rela, x 251]; 

2721, [ test 21 Zyl, = 2, X 22, Ll, 

Zylyy 2lyy Z2ly [test 24, = 22x Ip], 
Bee ee eal [ test eA Nas ae zl, |, 

ERE [test 23250, 2, X 2 2y!5]. 


If the calculation of these numbers is carried out in 
tabular form, like functions in one column and functions 
belonging to the same observation in one line, the 


IX. On the Statistical Theory of Heat Radiation. By 


Statistical Theory of Heat Radiation. 121 


47 sums wanted can be found by simple addition of the 
numbers in the 47 columns (omitting the third to the ninth 
columns,—whose sums are not wanted, and which may be 
written on a shéet by themselves ; as these seven numbers 
have to be combined by addition and subtraction, it may be 
most convenient to arrange them in columns corresponding 
to the several observations, to effect their combinations in 
these columns, and to transfer the values of /, and /, thus 
found to the main sheets). 
Worcester, Mass., U.S.A., eae ; 
January 1, 1909. ames f 


Prof. Harotp A. Witson, F.R.S., McGill University, 


Montreal * 


Te theory of the distribution of the energy in the 

spectrum of full radiation which we owe to Planck 
has recently been presented in a new and more general form 
by Sir J. Larmor {. In Planck’s theory the energy is taken 
to be emitted by “ resonators ” contained in the body which 
are supposed to only gain and lose energy by certain finite 
increments the magnitude of which is proportional to the 
frequency of vibration. On this view it is not absolutely 
necessary to regard the radiation itself as made up of finite 
elements ; but Hinstein and others have shown that Planck’s 
theory can beso interpreted. Larmor considers the radiation 
itself as made up of “elements of disturbance”? which are 
regarded as definite entities possessing energy, but the energy 
in an element can vary continuously. 

Larmor states that onhis view “it would be the limiting 
differential ratio of energy element to extent of cell that is 
somehow predetermined, but now without any implication 
that energy is itself constituted on an atomic basis.” I find 
that J.armor’s theory seems to require the radiation to be 
made up of finite elements of the same magnitude as those 
contemplated by Planck and Einstein. Tlis does not mean 
that energy itself has an atomic constitution, but it does 
appear to require some such sort of constitution for the 
radiation. N 

Larmor obtains the equation cO-=log( 1+ =} where ¢ 


denotes the energy per element of disturbance of a particular 
wave-length, n the number of such elements, and N the 


* Communicated by the Author. 
t Proc. Roy. Soc. A. vol. lxxxiii., 1909. 


1322 Prof. H. A. Wilson on the 


number of “cells” into which the ether is supposed divided 


for radiation of the wave-length under consideration. @ is 
a function of the temperature, and is the same for all the 
different sets of cells. 

To determine the relation between @ and the temperature 
(1) on the conventional absolute scale we have Boltzmann’s 
expression for the entropy S=‘log W, where W denotes the 
number of ways in which the sys stem can be arranged in its 
actual state. Hence 


dS = 110 W) =p Iog(14 si 


dn dn 


for the system consisting of the n elements distributed among 
N celis. If nis increased by unity, the increase of entropy is 
e/t, so that 


Hence i 


Larmor shows that Noc1/A3 and ex 1/X; so that after 
multiplying by 1/X to allow for the variation of dA we get, 
putting e=he/A, 


" he —1 
a or e Akt — 1) } 


where ea denotes the energy density per unit range of wave- 
length and c¢ the velocity of light. To get G we can make 
use of the value of e, for long wave-lenyths calculated by 
H. A. Lorentz and Jeans, viz., a= Barkt/A4. Hence 


CO rkt _ 8arkt | 
Nhe ok ES 


. : he —1 
so that 6=8rhe and e= mel ext ~1) 


which is Planck’s formula. 
H 


ee : 
In the formula ne= ey if we suppose e indefinitely 


diminished while X is ort constant, we get ne= Nkt, so that 
the energy per cell is At and is the same for all the sets of cells. 
This is merely equipartition of energy and corresponds with 
ex=S8rkt/A*. It seems therefore that ¢ cannot be made 


Statistical Theory of Heat Radiation. 123 


indefinitely small on Larmor’s theory any more than on 
Planck’s*. 

The total number of elements of energy per c.c. (1) is 
given by the equation 


© Pe ev 
a= Tee ixt—1) vin 


ag he 


Hence 
Merete he at 
H=160 (=) (14 5 BE et es. 


The series in brackets is equal to 1:20....=2' say. 
The total energy per c.c. is 


A8arak*t* 
a sore ne 
eh ” 
é ] 
where a=1+ = 7 += 5 +....=1°0823. 
Let € ee average energy per element so that 
€= = = 3kt ie 


Now 3kt/2 is the average translational energy of a molecule 
of a gas, and 


It appears, therefore, that the average energy per element 
of disturbance in the radiation is equal to 1°80 times the 
energy of a monatomic gas molecule. This result, it will be 
observed, is independent of the absolute values of the constants 
in Planck’s formula. 

The pressure (p) of the radiation is equal to H/3, so that 


Es 
a 


For a gas we have p= Wit if Y now denotes the number of 


molecules per c.c. Thus for a given pressure and tempe- 


rature a gas contains 0°90 times as many molecules per c.c. 
as full radiation contains elements of energy. ‘The elements 
of disturbance have on the average as much energy as if each 
possessed 5:4 degrees of freedom and equipartition held good. 
For a gas each molecule of which has six degrees of freedom 


* The value of « for any wave-length is of course given by e=Ac/A, 
using the value of # required by the observed values of e,. 


124 On the Statistical Theory of Heat Radiation. 
iE 
3 
per c.c. Also for an adiabatic expansion of such a gas 
pe*®=const. These two equations also hold for full radiation, 
which suggests that the elements of disturbance ought to 
have energy corresponding to six degrees of freedom instead 
of only 5°4, but the energy is not distributed among the 
elements in the same way as among the gas molecules. 
Consider the free expansion of full radiation from a volume 
v, toa volume v,.. The chance that an element is in v, when 
the volume is v, is v,/ve. Thus the chance that all the Nv, 


we have p= 5, where E denotes the total energy of the gas 


: a ake 
elements are in v is a) ; hence the increase of entropy 
v; 
5 . . BR ; . Vo . 
S,.—S, due to the free expansion is ke¥v, log rl, If vg- v, is 
1 
very small, say dv, this becomes k4dv=d8. Now 


7 see zi 


so that for an infinitesimal free expansion, if for the moment 
we regard ¢ as unaffected, we have 


Hence tHadv= "dr, 


or - € =:Bkt instead of €=3ht=, 


This makes € equal to the energy of six degrees of freedom, 
but the supposed infinitesimal tree expansion alters the tem- 
perature of the radiation by different infinitely small amounts 
forthe energy of different wave-lengths. Consequently it is 
not clear that even after only an infinitesimal free expansion 
the radiation can be regarded as having a definite temperature 
differing infinitely little from ¢. 

In the case of the gas we have in the same way for a free 
expansion 


dS=kWdv= “__ == ~ ______., 
t 3 t 


where m is the mass of a molecule and wu? the average square 
of the velocity of the molecules. Hence 3kt=3mu?, which 
gives the value of A due to Planck. The known equation 


Amount of Thorium in Sedimentary Rocks. 125 


ch 4 5 
—_— . yo oO xy .) ~ 
Ant = To e5y where 2,, is the wave-length for which e, has 
its maximum value, gives with the expression found for Y 


ih) Lea! 
965 N),)F 


Consequently the number of elements of disturbance per c.c. 
can be calculated from X,, without knowing the density of the 
energy. Since A,,t=0°294, we get approximately 


AWN =19°5 83, 


Thus at 2000° on the absolute scale there are 156x101 
elements per c.c. in full radiation according to the theory 
considered here. 


Montreal, April 13, 1910. 


3 £ ¥ 
, a 


X. The Amount of Thorium in Sedimentary Rocks.— 
I. Caleareous and Dolomitic Rocks. By J. Jouy, F.R.S.* 


| aie systematic determination of the amount of thorium 
in sedimentary rocks does not seem to have been 
hitherto attempted. Using a method already fully described 
by me (Phil. Mag., May and July 1909) I have recentlv 
measured the thorium content of calcareous and dolomitic 
rocks from various parts of the world and of various geolo- 
gical ages. The results are given below. 

In all cases, the rock after being brought to a coarse powder 
was treated with 100 ces. of HCl diluted to a bulk of 200 ces. 
with distilled water. A test applied to 500 ces. of the acid 
used showed no trace of thorium. After the first violent 
effervescence had ceased, the whole was heated for a couple 
of hours on the water-bath. The undissolved part was then 
filtered off, dried, and fused with about twice its weight of 
the usual fusion-mixture of the carbonates of sodium and 
potassium. The melt was then leached with water and 
acidified with sufficient acid rapidly stirred up with it. In 
most cases a clear or almost clear solution resulted, which 
could be added to the solution containing the soluble part of 
the rock. In a few cases, where the insoluble residue, 
obtained after treating the rock with HCl, was large, the 


* Communicated by the Author. 


GISB9 


126 Prof. J. Joly on the Amount of 


residue was, after fusion, divided into an acid and an alkaline 
solution; the former being alone added to the original acid 
solution, and the alkaline solution reserved for a separate 
test. 

It was thought best to determine the constant of the 
electroscope by an experiment in which a known amount of 
a standard solution of thorianite was added to one of the 
acid rock solutions which had been already tested for thorium. 
Throughout the experiments all the solutions were boiled in 
the same flask and brought to the same bulk. These con- 
ditions, as well as those of velocity of air-current &c., were 
preserved unchanged when finding the constant of the 
electroscope. It was found that when one cubic centimetre 
of astandard thorianite solution containing 6°977 x 10-4 gram 
thorium per ce. was added to the rock solution, the rate of 
discharge of the electroscope increased from a quite steady 
rate of 7°3 scale-divisions per hour to 50 scale-divisions per 
hour, a gain of 42°7 scale-divisions. Consequently one scale- 
division per hour increase corresponds to 1°63 x 10-° gram 
thorium, which is the required constant. A previous deter- 
mination of the constant of this electroscope under the same 
conditions, but effected by adding the thorianite to the solu- 
tion of a trachyte of the Andes, gave the constant as 
1°68x10-° gram. The first value was used throughout. 

The figures in the brackets, given in the table which 
follows, refer to the weight in grams of the amount of rock 
dealt with. 

It will be noticed that in most cases the quantity of 
thorium present is so small that it could not be certainly 
determined in the quantity of rock used. The major limit, 
in such cases, is obtained by dividing the constant of the 
electroscope by the weight of rock taken; it being assumed 
that a change in rate of discharge of one scale- division per hour 
is readable. ‘This assumption is certainly justifiable. 

It would appear from these results that thorium is not 
abstracted in any considerable degree from waters in which 
calcareous materials are formed. ‘This may be due to a 
process of organic selection among the dissolved materials. 
I have obtained, in the case of a sample of sea-water froin 
the Indian Ocean, 0°9x10-§ gram thorium per gram of 
water (Phil. Mag. July 1909). I£ this approximates to the 
average thorium content of sea-water, by far the greater 
part of the thorium accompanying a given amount of calcium 
salts in sea-water must be rejected in the organic processes 
attending the abstraction of the lime. 


Thorium in Sedimentary Rocks. 127 
TABLE. Tuorium. 
(gr. per grm. X 10-5), 

1. | Littorina Limestone (Pleistocene). Rhein Hesse. (45) o.......e. ccc ccceeccccceueceee 013 

Paemrmimulite Mamesmome, MOC. 2 (SU) sth iseceeecde ccc sctceedeccsetdescd descccsccce Aeatee! 0-08 

EE MEAT TNIGE SO MAU LOE: F(R) nice atin'vaideas's ated sce's+'sie canes siesbinenvese ve even less than | 0:06 

aemernper Chalk. 2Oo, Antrim.” \(50).) ai22is.620 i. cendees esse cee BIG S viwee ems less than | 0:03 

inonver Clinic.) Esle Of Wirt) (G4)s \ ecscnpaai atts dacceescsncesscccecsoecevece less than | 0:05 

6.| Earthy Limestone (Upper Cretaceous). Were, Westphalia. (30) .................. U-22 

7.| Dolomitic Flagstone (Portlandian). Embeckhausen, Hannover. (30) less than | 0-05 

Pete. | 'alean Portiand., Dorspt. (DO), Ssescdsccscctsccccecescecceccedsocee less than | 0:03 

9.| Lithographic Slate (Kimmeridgian). Solenhofen, Bavaria. (39) ...... less than | 0:05 
10. | Limestone (Jura). Hohenstein, Saxony. (30) ..............ccccccesecece nee less than | 0:05 
ft), |)amestone (Jura). Schnaithemm, Ulm, (30) .....2..........ceceeeneceee . less than | 0-05 
PEN ATCA MI fora. cde Sana duan ec eradesaxelnwe cee csenecaneescccecnss res less than | 0-05 
13. | Limestone (Malm). Untergletcher, Grindelwald. (40) ...........000.... less than | 0-yz 
14. | Limestone (Dogger). Sandfirn, Todi. (40) ......... Aah SAcRSEE SEM mee less than | 9-9 4 
foveveute Marble (Trias), Osarrara, Italy. (30) .22..6..-cecck.csscsecescecee: less than | 9.95 
eee petunonenkalio \EVeidolbere. (UY ji icecrcscesscwcecesedenphoves es dadegeessscceees less than | 9.93 
17. | Limestone (Zechstein). Tettenborn, Hartz. (380)  ......... eee. less than | 9.95 
18. | Limestone (Upper Carboniferous). Co. Kildare. (30) ..........00.000.. less than 0-05 
19. | Limestone (Carboniferous). Armagh, Ireland. (22°5) oo... less than | 9. 79 
20. | Oolite (Carboniferous). Ballina, Co. Mayo. (50)... eee. less than | 9.92 
21.| Limestone (Carboniferous). Nuttlar, Westphalia. (39) ...........0000... less than 0-05 
22. | Limestone (Carboniferous). Dusseldorf. (30) ..................cceccesesees less than 0-05 
23. | Limestone (Lower Carboniferous), Co. Kildare. (30) 22.0.0... cece cee ecececeuees 0-14 
Ptewer Uimestone shale. Wo, Kildare: (G0)? soiciicii occ. oe ccc dee calevecuveccdacecces 0°33 
25.| Grey Marble (Upper Devonian). Namur, Belgium. (80) ...2........... less than 0-05 
26. | Black Marble (Devonian). Shuppach, Nassau. (GU) ate ee eee eee less than 0-05 
za neiwmite (Doyoniam). — Gerolstein, Wifel. (GO) 2....c25cccc.ece wees cceccesedacscdceces O16 
28. | Favosites Limestone (Devonian). Torquay, Devon. (30) ............... less than | 9.5 
29. | Coral Limestone (Upper Silurian). Aymestry. (85) ............. 2 ee Pe less than | p.95 : 
30. | Porites Limestone (Middle Silurian). Wenlock. (35) .................. less than 0°05 : 
31.| White Marble (Lower Silurian). Boda, Norway. (380) ................4. less than | p.95 | 
32. | Cheirurus Limestone (Ordovician). Chair of Kildare. (40) ............. cece. 0-05 . 
33.| White Marble (Archzan). Pentelicon, Greece. (80) ..............000000. less than | 9.95 | 
3.| Dolomite. Fichtel Erzgebirge; Sayda, Saxony. (30) ............eeeeeenee less than 0-05 


In the case of two rocks dealt with (Nos. 6 and 24) there 
were at once very considerable insoluble residues, and a 
thorium content greater than was generally indicated. It 
seemed of interest to find if the thorium was in these cases 
in the soluble or in the insoluble part of the rock. The 
soluble part of 42 grams of No. 6 was tested separately and 
found to contain 5°3 x 10-° gram thorium. The total thorium 
in 42 gram is 9°1x10-°. Hence there is no special concen- 
tration of the thorium in the insoluble part, which in this 
case was found to be very nearly 40 per cent. by weight of 
the rock. In the case of No, 24 the soluble part of 50 vrams 


128 Dr. W. H. Eccles on an Oscillation Detector 


was found to contain 5°05x10-° gram thorium. As the 
entire 50 grams should contain 16°3x10-° gram thorium, 
and the soluble part was found to be just 71 per cent. of the 
rock, there is here a considerable concentration in the 
insoluble residue. 

It may be remarked as regards the geological significance 
of these results. that the caleareous rocks have been estimated 
as constituting about 5 per cent. of the bulk of the total 
sedimentaries upon the land*, and although somewhat higher 
estimates have been made, they certainly constitute a small 
fraction of the sedimentaries. 

Soils derived from such rocks consist largely of the in- 
soluble residues, and hence the influence of the surface 
materials in ionizing the atmosphere over calcareous districts 
cannot be directly inferred from such results as the fore- 
going. 

Geological Laboratory, Trinity College, Dublin, 

May 18, 1910. 


a 


Syd 


\ XI. On an Oscillation Detector actuated solely by Resistance- 
“temperature Variations. By W. H. Eccuss, ).Se.t 


N a recent communication to the Physical Society { the 
properties of a type of iron-oxide coherer were discussed. 

‘The paper described experiments on coherers made by dipping 
a slightly oxidised iron wire into clean mercury, or by 
pressing a fine iron wire against a thinly oxidized iron plate, 
and the results were discussed mathematically. It was shown 
that in the case of the iron point and oxidized plate the 
whole of the experiments could be explained qualitatively on 
the assumption that the only electrical phenomena at play 
were the Joule ettect and the resistance-temperature changes 
in the small mass of oxide between the metal electrodes. 
The hypothesis that was put forward must be summarized 
here. Let p be the resistance of that part of the detector 
where the current flow is so constricted that the Joule effect 
produces rise of temperature, and 7 the resistance of the 
remainder of the circuit. The resistance p is usually localized 
at the contact of the conductors that form the detector, and 
it varies with the temperature of the minute mass of matter 
at the contact; let « be the coefficient of decrease of resistance 
with rise of temperature. The resistance r includes that of 
the bulk of the substances forming the detector, the leads 


* Van Hise, ‘ Treatise on Metamorphism,’ p. 940. 

+ Communicated by the Physical Society : read May 27, 1910. 

{ W. H. Eccles, ‘‘ On Coherers,” Phil. Mag. June 1910; Proc. Phys. 
Soe. vol, xxii. 1910. 


actuated by Resistance-temperature Variations. 129 


and the telephone; it is supposed constant. Let ¢ be the 
current when the electromotive force applied to the detector 
ise. Assume that the rate of loss of heat from the warmed 
contact is m@, where @ is the temperature of the contact 
above that of the surroundings. Then it was shown that in 


the steady state 
Syl dbeiEkiBar | ay 
AO a Me Gituk o 


where py is the resistance of the contact when cold and g is a 
constant. If the curve e=p,c/(1+ ac?) be plotted with ¢ as 
abscissee and ¢ as ordinates, it is seen to rise from the origin of 
coordinates with an increasing gradient till at a definite value 
of € it becomes vertical. Then asc increases, the curve bends 
towards the axis of ¢ and approaches it asymptotically. 
Along this latter part of the curve, increasing current 1s 
associated with decreasing electromotive foree—an unstable 
state of affairs. Any conductor possessing a negative tem- 
perature-coefficient of resistance exhibits these properties. 
In such conductors an increase of current produces, in ac- 
cordance with Joule’s law, an increase of temperature, and 
consequently a diminution of the resistance. The curve 
shows that after a certain stage is passed the diminution of 
the resistance which accompanies increasing current is great 
enough to lead to a catastrophe. Stability can, however, be 
obtained by introducing into the circuit of the variable re- 
sistance p a sufficiently large constant resistance 7. The 
phenomenon resembles that of the electricarc. The unstable 
portion of the above curve corresponds, in fact, to the “ falling 
characteristic ” of the are. 

If the resistance 7 is large the e, ¢ curve has a positive 
gradient throughout, and the gradient has a maximum at 
some value of «. Near this point the contact is found to 
work best as a detector of oscillations. The hypothesis put 
forward by the author supposes that a train of oscillations, 
by yielding its energy as heat to the contact, raises the 
temperature there and disturbs the existing equilibrium of 
current and voltage. The dissipation of a train of oscillations 
is accomplished in a time of the order of a fifty-thousandth 
of a second; the ensuing fluctuation of the current causes 
the sound in the telephone. In the paper cited, the energy w 
given to the telephone circuit was shown to be connected 
with the energy W delivered by the train of waves by the 
equation 


See Wa)e SL eo vet BI) 
where m and a are constants for any fixed value of the 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. K 


130 Dr. W. H. Eecles on an Oscillation Detector 


current. This theoretical conclusion agreed well with the 
experiments on an iron oxide coherer. The object of the 
present paper is to show that the above hypothesis holds 
good for a very different type of detector. | 

All the well-known forms of “ contact” detector consist of 
a contact between two substances that stand well apart in 
the thermoelectric series; and the thermoelectric force plays 
a very important part in their operation. They are usually 
classed as “rectifying detectors” to distinguish them from 
such detectors as the coherers. In seeking a “ rectifying 
detector” that would illustrate the above hypothesis, sub- 
stances that stand far apart in the thermoelectric series 
must be avoided, for such substances would introduce Peltier 
effects that would tend to mask the resistance-temperature 
phenomena we wish to isolate. Hven in the iron: iron- 
oxide: iron coherer it is possible that thermoelectric forces 
arise owing to unequal heating of the two iron to iron-oxide 
junctions. It occurred to me, therefore, to construct a 
detector out of one substance only. Accordingly a search 
was made for a substance possessing high, but not too high, 
resistivity, with large negative temperature-coefficient. 
Several such substances were found among the native crys- 
talline oxides and sulphides. Most of these are selotropic, 
and must on that account be avoided. Fortunately galena, 
the native sulphide of lead, has fairly high resistivity, a very 
large negative resistance coefficient, and crystallizes in the 
cubic system. A galena-galena detector was therefore 
constructed. Two pieces of galena cut from the same 
crystal were embedded in solder, mounted in a clamp, and 
brought into gentle contact. It was put into a wireless 
telegraph receiving set and was found to yield excellent 
signals when a current of proper magnitude was passed 
through it. . 

A tew preliminary experiments showed that it was not 
easy to find two pieces of galena, which when placed together 
and submitted to positive and negative voltages in turn, 
exhibited perfect symmetry. The asymmetry is always 
small, and is probably due to the rise of thermoelectric forces 
at the contact when it becomes heated by the steady current, 
and these forces are probably caused by slight variations in 
the chemical composition of the galena from point to point 
in the crystal. When a contact that gives symmetrical 
voltage-current curves is found, it constitutes a detector that 
operates only when a current is passing through it. The 
efficiency of the detector is the same for each direction of the 


Oe eS ee 


actuated by Resistance-temperature Variations. 131 


current, and appears to be practically independent of the shape 
of the galena surfaces at the contact. 

The experiments now to be described were made on selected 
galena-galena combinations in precisely the same way as the 
earlier experiments on the iron-oxide detectors. The diagram 
of the apparatus employed is shown in fig. 1 of the previous 
paper (Phil. Mag. June 1910, p. 872). The mutual in- 
ductance between L and L’/ was 2500 cm. in obtaining the 
results described below. 


0 0-2 0-4 0-6 0-8 1:0 1-2 14 Volt 


The detector was placed at D and submitted to an electro- 
motive force from P,, which was varied in steps from zero 
to about one volt positive or negative. At each step an 
observation was made of the steady current through the 
detector, and of the intensity of the sound produced in the 
telephone by electrical oscillations of amplitude fixed by the 
position of the jockey on P;. The steady current observation 
was made by means of a shunted galvanometer kept connected 
in series with the telephone. The intensity of the sound was 
measured by balancing the sound from the detector against 
that from the interrupter and the potential-divider P;. Thus 

K 2 


132 Dr. W. H. Eccles on an Oscillation Detector 


were obtained the curves of figs. land 2. Fig. 3 was obtained 
by applying oscillations of various amplitudes to the detector 


Fig, 2. 


107° Watt 


while it is traversed by a suitable steady current, and 
measuring the intensities of the resulting sounds ; and there- 
fore expresses the relation between the energy given to 
the detector and that appearing in the telephone circuit. 
The curves given here are selected out of a number as repre- 
sentative ones. The absolute values of the energy quantities 
involved in the results given in this paper were determined 
in the same way as in the former paper, and are therefore 


actuated by Resistance-temperature Variations. 133 


subject to the same errors ; the errors in the absolute values 
of the energy delivered to the telephone circuit may be 
large. 

The most difficult observations to make were the steady 
current ones. The current-voltage curves obtained with 
increasing currents are very different from those got with 
decreasing currents unless certain precautions are observed. 
Usually a decreasing-current curve lies above an increasing- 
current curve. The difference between the curves was found 
to depend greatly on the time allowed for a set of measure- 
ments. If, with decreasing currents, the electromotive force 
be held constant, the galvanometer reading slowly diminishes, 
till in about half an hour it has fallen as a rule to the value 
it would have at the same voltage with increasing currents. 
On the other hand, if the voltage be raised suddenly from 
zero to a fairly large value, the reading of the galvanometer 
increases as time elapses, at first quickly and then slowly, 
rising perhaps fifty per cent. in five minutes, and asympto- 
tically approaching a limit. All these things are due to the 
temperature at the contact lagging behind the changes of 
current. This may be understood as follows :—At any stated 
value of the steady current the mass of galena surrounding 
the heated contact must be cooler when the current is being 
increased than when it is being decreased—for the reason 
that in the former case the conductor has just previously 
been carrying smaller currents, in the latter case larger 
currents, than the particular current considered. The rate 
of loss of heat from the heated matter at the contact to its 
surroundings is thus greater on the rising curve than on the 
falling curve, and, in consequence, at corresponding points 
on the two curves the resistance is higher in the former than 
in the latter case. This temperature lag may be expected to 
be more pronounced in detectors made wholly of substances 
~ of low thermal conductivity than in detectors consisting of a 
very thin film of oxide between metal electrodes. The phe- 
nomenon is very prominent in this new galena detector 
though it escaped observation in the iron-oxide detectors, and 
therefore the former may be expected to work well as a 
rectifier of alternating currents of moderate frequency. 

Once the slow movements of heat through the galena had 
been observed the difficulties were overcome by allowing 
time for thermal equilibrium to be attained before the final 
galvanometer reading was taken. At places near the point 
of inflexion of the steady current curve the time necessary 
may be several minutes. 

Each setting of the two pieces of galena gives a different 


134 On an Oscillation Detector. 


contact resistance and therefore a different curve. Since 
very slight accidental vibrations can cause relative motion 
of the pieces of galena, it is almost impossible to perform a 
series of measurements yielding sufficient data for all the 
curves of figs. 1, 2, 3, with full confidence that the contact 
has not varied. 

Curve a of fig. 1 follows the equation. 


a. SS Os A 
eE= 14 2°95 x 10°? + 40x 10°) c . : (3) 


with fair accuracy ; curve ) obtained with another setting 
of the crystals has the equation 


( 4° x 104 
tye 


ah 
14 83x 10° + Va TIA 10° Je. . = fe (4) 


The curve of fig. 2 was obtained we the same setting of 
the crystal that gave curve a of fig. 1. It was shown in "the 
previous paper that the ordinates of this curve should be 
proportional to m the coefficient of W in equation (2) above, 
and that m contained the gradient of the steady current 
eurve as its principal factor. A comparison of figs. 1 and 2 
confirm the deduction. The line of fig. 3 has the equation 


w=0-077 (W—0-4 x 10°), 


where W is the power in watts given to the detector in the 
form of oscillations, and w the power transformed. by the 
detector and passed to the telephone circuit. 

The galena detector here investigated proves very good 
in practical wireless telegraphy. It has shown itself to be 
better than any coherer * known to me. This superiority is, 
according to the hypothesis here advocated, to be ascribed 
chiefly to the large negative temperature-coefficient of re- 
sistivity of galena. By direct measurement of a cube of 
galena clamped between pieces of tinfoil the resistance was 
found to fall from 0°33 ohm at 12° C. to 0:10 ohm at 99° C. 
—a negative coefficient of 0-0079 per degree centigrade. 
Pyrites has a coefficient about 0:006. Iron oxide has a 
coefficient somewhat lower than this last. 

The cost of a portion of the apparatus used in these 
experiments was defrayed out of a grant from funds at the 
disposal of the Royal Society. 


* But not so good as certain ‘‘ thermoelectric ” detectors. 


[ 135 ] 


XII. On the Theory of Surface Forces —V. Thermodynamics 
of the Capillary Layer between the Homogeneous Phases of 
the Liquid and the Vapour. By G. BAKKER™. 


§S 1. Some dejinitions and mathematical signifiéations. 


HE surface-tension of Laplace may be conceived in two 
ways. Firstly, as the increase of the available energy, 
when a part of the liquid- and vapour-phase is converted 
isothermally into the matter of the capillary layer, in con- 
sequence of which the surface of the latter has increased by 
a unit of surface. The contributions of both homogeneous 
phases to the formation of the capillary layer are here com- 
pletely determined, as we assume that liquid and vapour 
are present in abundance, while the influence of the walls of 
the vessel is not considered. We imagine the process to be 
very slow. 

What we call a unit of surface for a plane capillary 
layer is evident. For a spherical capillary layer we deter- 
mine it as follows. The spherical capillary layer lies between 
two concentric spheres. The radius of the smaller sphere 
we call R, and that of the larger one R,. We take as 
the “ surface” of the capillary layer that of the sphere the 
R,+R, 

) . 


=! 


radius of which has the value R= 


The capillary 


energy that belongs to one unit of this spherical surface 
we call the capillary energy or constant of Laplace. If the 
curvature of the capillary layer is small (e. g. measurable) 
itis unnecessary to distinguish R,, Ry, and R. However, 
if the curvature is very large (R,—R, and R may be of the 
same order of magnitude+) we conceive a great many parts 
of spherical capillary layers of equal curvature, the collective 
surface of the resp. spherical surfaces with radius R being 
unity. The surface energy or the available energy, pro- 
duced at the formation of the capillary layer per unit of 
surface (radius R) we represent by H. [Fuchs proposed 
another conception for the surface-tension of Laplace. He 
perceived that the cohesion in directions respectively perpen- 
dicular and parallel to the surface of the capillary layer has 
not the same value. The difference between these cohesions 
he considered as the cause of the surface-tension. He has. 
however in this way not calculated the surface energy f. 


* Communicated by the Author. 
+ G. Bakker, Phil. Mag. March 1909, p. 546. 
t K. Fuchs, Wiener Ber. xeviii. 2 Nov. 1889. 


y 


A 


136 Dr. G. Bakker on the 


If we consider the thermic (kinetic) pressure as inde- 
pendent of the direction, we may assume the difference 
between the hydrostatic pressures in a point of the capillary 
layer to be equal to the difference of the cohesions both in 
directions resp. perpendicular and parallel to the capillary 
layer. If py and py are the pressures in a point of the 
capillary layer respectively in directions perpendicular and 
parallel to the surface of the capillary layer, I have called 
Py—Ppv the departure from the law of Pascal at the considered 
point. The capillary constant of Laplace thus becomes the 
integral of the expression py—pa, or, if py and py represent 
average values, we have for a plane capillary layer 

H=(p,—p)é, 

in which € represents the thickness of the capillary layer. 
(For a plane capillary layer p,=py=ordinary vapour-pres- 
sure.) By integration of the expression py—pr Hulshof and 
Bakker calculated the capillary constant of Laplace. Bakker 
deduced the complete expression of Rayleigh, and has given 
also an elementary proof of the exactness of the conception 
of Fuchs*. As p, at a definite temperature (about +T,) has 
the value zero+, we have for that temperature: 


capillary constant 
vapour pressure : 


we 


We may thus, at least for the mentioned temperature, con- 
sider this quotient as the value of the thickness of the plane 
capillary layer. 

If we consider such a large surface of the body, that the 
total mass of the capillary layer is unity, we call its surface 
S and consider S€=v as the specific volume of the capillary 
layer if it is plane. For the spherical capillary layer (see 
above) the thickness is expressed by (R,—R,); we represent 
it again by ¢ It will easily be understood that in this case: 


poate 8: 
v=SCO+ — Be : 
out 
For the cylindric capillary layer on the contrary we have 
: se : f Ri+k 
again v=S¢, in which § belongs to the radius R= ae 


* H. Hulshof, Koninkl. Akad. v. Wetensch. at Amsterdam, 29 Jan. 1900. 
G. Bakker, Zeitschrift f. phys. Chem. xxxiii. p. 499 (1900); and Phil. 
Mag. for Dec. 1906, pp. 565 & 569. See Rayleigh, “On the Theory of 
Surface Forces II.,” Phil. Mag. Feb. 1892, formula (22). 

+ G. Bakker, Phil. M ag. Oct. 1907, p. 522, and Zertschr. f. phys. Chem. 
i. 1905, p. 359, 


Theory of Surface Forces. Eat 


The pressures in the homogeneous phases we call resp. 
pr and p, or py and py. The specific volume of these phases 
we represent by v, and vp. 

Upon the whole we have the following significations : 


¢€ =Thickness of the capillary layer ; dh=differential of &, or 
2 
dh=€. 


; a 

S = Surface of the plane capillary layer per unit of mass. 

R, and R,= Radii of the two spheres which limit the spherical 
capillary layer; R,> R). 


pe at Re 
5 ; 


S = Collective surface of the spheres with 
radius R, the total mass of the spherical capillary 
layers being unity. We call 8 the surface of the 
spherical capillary layer per unit of mass. 

v = Specific volume of the capillary layer or total volume of 
the capillary layers of equal curvature per unit of mass. 


| : 

= is Mean value of the density of the capillary layer. 

pi ovr pr = Hydrostatic pressure of the homogeneous liquid 
phase. 

Pz OF p» = Hydrostatic pressure of the homogeneous vapour 
phase. 


v= Spec. volume of the liquid phase ; p,= its density. 
v= Spec. volume of the vapour phase ; pp= its density. 
je ee 
») 9 


a 


Vy- Yy=U;~P = p'= pressure of the theoretical 
isotherm. 


1G ee Pe P1141, 
p=) p'dv = Mean value ok the pressuxe = —-— — 5 
u ) 


1 
y = latent heat of vaporization. 

r= 1— pu. 

e, and e,)= resp. the energy of the liquid and vapour phase. 
Thus : 7;== €)— &- 

m, and 7.= resp. the entropy of the liquid and vapour phase. 
Thus : r=T(m.—™)- 

py and p.= resp. the thermodynamic potentials. 

¢ and = resp. the energy and entropy of the capillary layer 
per unit of mass. 


Vg— V7 


' 2H 
R, =Radius of the equation of Kelvin: pi-p2= hia 


138 Dr. G. Bakker on the 


H = Available energy, total departure of the law of Pascal 
per unit of surface, constant of Laplace or surface- 
tension. 

px = Hydrostatical pressure in a point of the capillary layer 
in a direction perpendicular to its surface. 

Pr = Hydrost. pressure in a direction parallel to the surface 
of the capillary layer. 
~ 29 ey) 2 

eee Cah: p= -| prahs pyle al pth? De = , pydh’. 

oo oh, ov oN 
§ 2. The radius of curvature of a capillary layer and 
the equation of Kelvin. 

The thickness of a capillary layer that limits a spherical 
liquid mass of measurable curvature may be neglected with 
respect to the radius of the liquid mass, and it is therefore 
indifferent whether we consider R,, Rg, or R. 

Quite otherwise is it on the contrary if the value of R 
lies e. g. between one micron and the minimal value of the 
radius of the spherical liquid mass. In a preceding article 
I found that the minimum value of the radius of a liquid 
drop is of the same order of magnitude as the thickness of the 
capillary layer *. 

If R, and Ry are resp. the radii of the spheres which 
limit the considered spherical capillary layer, we put 


R=——>— and call R the radius of the drop. The matter 


within the sphere of radius RK, should be considered as a 
liquid. If we eall », the pressure in the liquid mass of the 
drop, p, the pressure of the vapour which limits the drop, 
H the surface-tension, and Rx a value between R, and Re, 
Kelvin found as is known: i 
2 
7 I) . . . . : . 
Pi Pr Rx (1) 
Meanwhile R, has for very little drops a rather complicated 

signification. If for a point in the capillary layer we call 
px the pressure in a direction perpendicular to the surface 
(radial), and p, the pressure in a direction parallel to the 
surface, we have: 


dpy 2 (py—pr) ¢ 
fie aE PieePal 9 sa 


in which dh represents the differential of the normal to the 


* Phil. Mag. March 1909, p. 346. 
t G, Bakker, Phil. Mag. April 1908, p. 422, formula (17). 


f 


Theory of Surface Forces. 139 


surface of the capillary layer, while R’ signifies the radius 
ot the sphere we construct through the Poceaterod point 
ore to the surface of the little dr op. By integration 


of (2) I found: 
ai, Geel )dh 


Pia—Pr= (3) 


The quantity Rx of Kelvin is at curvatures not too strong* 
consequently ne by: 


Di EG i —Ppr)dh | 
ran (px Balle {y uae 
Generally Rx is not identical with R= fee. ; 


« We now wish to bring Rx and KR into connexion with 
each other. For that reason we consider the equilibrium of 
the spherical capillary layer under the influence of the pres- 
sures resp. of the liquid (inside R,) and of the vapour around 
the little drop in the same way as the hemispheres in the 
celebrated experiment of Otto von Guericke on the atmo- 
spherical pressure. If the thickness of the capillary layer is 
represented by &, and the pressure parallel to the surface for 
a point of the capillary layer is denoted by p,, we find as the 
condition for the equilibrium: 


m(R—-36)?—7(R+36)"p,= —20 “pgRidh 
el 
—2r| p(R-+Idh. 2. (4) 
el 


As the thickness of a capillary layer only amounts to few 
millimicrons, we may, in the case when R is of the order of a 
micron or larger, neglect the terms with ¢?/R?, and have: 


a¢ ici afeae a 


* The surface-tension being the integral of the departure from Pascal’s 
law with respect to the volume elements, and not with respect to dh. 


Pi p= 


EB ie prdh? is represented by pyr'¢’, the complete equation is 


2 2 ; 
(m—pe)(1+ Zhx2) = ee — py ee where p= PEP : 


140 Dr. G. Bakker on the 


If we neglect the quantities of the order = with respect 


o> 


to unity, and put fen iy =p, we can conclude from (3) and (4): 


Be B.—(p—p,): (ox—ta) es 


In this consideration we might express the surface-tension 
of Laplace by the integral: 


H={" ( Ps = py adh. 
1 


However, if we wish to be perfectly exact, we are to 
determine what is to be understood by surface and unity of 
surface of a curved capillary layer, and it is self-evident to 
point out the sphere of radius R as ‘ surface of the capillary 
layer.” The capillary energy H depends consequently at very 
strong curvature on so many spherical capillary layers of 
equal curvature that the mentioned surfaces have a total 
surface of unity. Let us call the surface per unit of mass 
of the capillary layer, S. 

Through each point of the spherical capillary layer we 
imagine a sphere, concentric to the adequate little drops. 
We call points, for which the R of these spheres in the 
different capillary layers has the same value, homologous 
ones. If the totality of the surfaces of these spheres per 
unit of mass is 8’, and R’ the corresponding radius, we have: 


S/ $ 3, =(R, +h)? . hry 


if 8, represents the total surface of the spheres with radius 
R,. If we new consider the capillary energy as the volume 
integral of the departures from the law of “Pascal, we have 
consequently: 


HS=(" S'(ps—ps)dh. het 
el 


If we put R’=R, +A, then 8S’: S,=(R,+A)?: R,? or: 


2h h? :) 


chee 
="P1 (14+ p+ 7 Re 


and the capillary energy per unit of mass becomes: 


HS=8, |’ (ps—pa)dh + 7 an (ps—ps)dl? + ae), (px—pa)d 
(7) 


Theory of Surface Forces. 141 
If we put: 


2 » 2 
( pydl? =p,'C’, pydh? = py F?, ( prdlb=p,,'C° and 
el el ew 1 


1G, 
J pal? =pilC, 
the expression for H becomes (because 8, Se | 
Re Re 
H=(ps—Pr)o— = {Pa —Px—( ps! —pal)} 
3 Se eee aa X Leta: ay . nae ! 
+io{ ere mes = ce ei \ No ea 


‘ iL 
For the value R=1 micron San (about) for water 


at ordinary temperature, and so Fs =4.10-% Mo @ isla 


quantity of the order py—pz, we have thus for values of R 
to the order of the wave-length of light: 


ay haan 3 
The equation of Kelvin gives thus: 


eke Ger? 
wee eo es bald _ oe Ba gy 


As Rx has a complicate signification, we wish to express 
Pi Po by means of a formula, where Rei is replaced by 


R.{R= eh 


el 


To this end we put the equation (2) into the form: 
(Ri +h)dpy=—2(py—pr)dh, . . . (2a) 


and integrate with respect to 4 between 0 and €¢. In this 
way we find: 


+ py—2pr 
PITPo= (ee ENG, Ooms eee (11) 
In this equation p= PitPe and py and pr are resp. the 


average values of px and pr; 7. ¢.: 


1 1 
a) pxdh and al prdh. 
1 1 


142 - Dr. G. Bakker on the 


For small curvatures we may put P=p,, and (11) becomes: 


whereas (10) in this case changes into: 


2¢ Ps Pode 
Fue 


Pr a ee 


Really we may put in the latter case R=R,. 


§ 3. Latent heat of vaporization and surface-tension. 


We conceive such a number of spherical capillary layers 
of equal curvature that they form together a unit of mass. 
We assume these capillary layers to “be parts of spherical 
eapillary layers that limit little drops of liquid of equal R. 
If the quantities which are considered refer to the homo- 
geneous liquid phase, we denote them by the index 1, whereas 
we take for the homogeneous vaporous phase the index 2. 
If the thermodynamic potential is represented by pw, we 
have as the condition for the equilibrium of the capillary 
layer : 

yey Or E—Tyytpywi=ea—Tyet+porr, . (12) 


If v signifies the latent heat of vaporization, we have : 
r=T'(n2—7;) = €) — €] + Polo—p 10}. * a ° . (13) 


The matter of the equally curved capillary layers is deter- 
mined by two parameters. For one of these parameters we 
take the temperature, while we leave the second provisionally 
undetermined. We differentiate while the second parameter 
remains constant and get : 


dr dey de, dv, dv, dps fe dp, (14) 


Maat at teat Pat tap OO aT: 


If we denote the spec. heat at constant second parameter by 
c, we have: 


_ dey dv, dey , dry 
=p t Piggy? and @= Gp +P ap 


consequently : 


d dps dj 
a= = (g2— e+ == (On a ° ° 4 (15) 


Theory of Surface Forces. 143 
On the other hand from 


an follows a = pear See a. GLO) 
(15) and (16) give in this way : 
dp: l 
- =UV2 =o eral Ci a A, - ‘ < : (1 7) 


In the particular case when the curvature is zero p.=p, 
and we get from (17) the known equation : 
ce ane dp, 
Ug — U1 a, al? 
in which p, represents the ordinary vapour pressure. 
The equation (17) may be written in this way : 
2r dp, dp, feu he ot) | 
(ev) af T at waar dry * ae) 
After Kelvin we have further: 
2H 
ay R aha 


K 


(19) 


If we differentiate at constant Rx and take R, in the 
equation (18) as second parameter we find from (19): 
apy apa. Oa EL 


Me ae a 7 OY 


and (18) becomes : 
r=(v2—-0,)T 


dp 0+ v2 pd 
Are ee dl 


in which : 
2 Prr ps 0 Pir po 
p= D) ae 9 ° 
For Rg=, i. e. if liquid and vapour are separated by a 
plane capillary layer, p,=p,=p, and the last member of (21) 
disappears, and we get the known equation : 


dp 
| r= (v2—V}) Tore 
If p' represents the pressure of the theoretical isotherm, 

we put 

p= : 'dv 

l Ug — Vy 1 s ( 
or as 2 

P22 Pry =| pd», 

1 

we have Po2— Pir _ 


Cg UE 


144 Dr. G. Bakker on the 


We can easily prove the identity : 


V1 + V2 Pi P2 
Vo Vy 2 ; 


p—p= 
Consequently, if v,—v; is replaced by wu: 
pu pu= (vy +r») iz . 
By substitution in aus : 
If a is the coefficient of the art expression for the 


molecular pressure ap? and the thermic pressure of the form 
Tf (v), I have found : 


pila 
= (a—" Tt) (pi —p2)*. 
In the case when the equation of state is of the form: 
> a 
p=Ty (v)— ye? 


where a represents a temperature function, we have thus: 


da dy Up + Ve r dH 
(a- 18) p= 1 va Re (H- Tar ): 


The identity : _ ty) +t, Ppi— po 

DP aha 

gives by differentiation with respect to T, with second para- 
meter constant : 


d(p—p) UT Ue d — Pe 4 Pi Pe d (33 


‘aT i Cg dT a 2 dT 


*), (23) 
(23) and (18) consequently give in general : 


Pio ep ae gone) 
u Tat BR, ae Vo— U7} 


These relations we shall use in § 4. 


(24) 


§ 4. The equation of energy of the capillary layer. 


If « and 1—z are resp. the quantities of the liquid and 
vapour, which have formed the capillary layer without 


* Zeitschrift f. phys. Chemie, xii. p. 283 (1893). 


Theory of Surface Forces. 145 


changing volume and mass, and »v the specific volume of the 
capillary layer, we have: 
ee me a 1 gp PtPa Ps 
P1 P2 Pi P2 Pi \P2 
Per unit of mass of the capillary layer the available 
energy, which the homogeneous phases have engendered, 
becomes : 


dees calc gs —pyv;) - Bee E25, — PxV2) 


Pi—P2 P1i—P2 
= fy + rs + a a mar eM . 
P1— P2 Ug U1 


As we mentioned above we have: 


| ae : ("pa =i? 
l 


UVg— V1 Ug == UT iy) 


So the last expression becomes 


anv Papp 
Pi—P2 
If, consequently, the surface of the capillary layer per 
unity of mass is 8, the available energy of the capillary 
layer is given by : 


eee ee eae ESS sis! 26) 
) Pree Pe 
For, what we usually call the capillary energy per unity 
of mass and represent by HS is the difference between e—T 
and the expression (25). Further we have: 


fy ren dienes tee ac! (AO) 


Fe Side 62 om +12 PY, F Pore 
a a 2 £ 2 te 9 . 


After a simple deduction (26) becomes therefore : 
e— 15% 1 (g— BF) +p(o— "E*)=Hs*. (27) 


For a plane capillary layer p becomes the ordinary vapour 
pressure, and we find the equation for this case, as I have 
found already in a preceding paper f. 


] 


- ( : w ; 
Me dee ye a i p'dv, where p’ denotes the pressure of the theoretical 
2 1 


isotherm (see above). 
+ Ann. d. Phys. xvii. p. 492 (1905). 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910 L 


146 Dr. G. Bakker on the 


A second thermodynamic relation we find in the following 
manner. If we evaporate a quantity of the liquid phase 
giveu by the expression 


i alt at at (see above), 
P1— P2 
we get precisely the complex of the two homogeneous 
phases, which can form unit of mass of the capillary layer. 
The corresponding heat of vaporization is expressed by 


EPiPs P2 7 eee, 

RPE y= (v—u) 
If we further convert the complex into the matter of the 
capillary layer, we must still add a quantity of heat per unit of 


dH 


mass, expressed by —T a S. We have therefore: 
a boomy 
2-h— —S ar + (0-1) a 


In the same manner we have : 


dH ? 
m—1= 8.6 4 (4-0) 4. 


Hence: 
y— DS 8S 4 (v— OE) F Mee i) 


2 aT 2 
where u=v,—v, represents the difference between the specific 
volumes of the liquid and vapour. 
From the equations (27) and (28) we can now deduce the 
equation of energy in the following way: by differentiation 


of (27) we find: 
igs 2 (de, —Tde) ae —(n— ns) aT 
ie at p(o- 2" - “2)\ =HdS+SdH. . (29) 
By substitution of the expression (28) for 7— nom , and 
as 


: de—Tdn,= — pdr, and déy — Td, = — podr. : 
de, —Tdn—Hd8S = —§ pydv, —§ poly + G er 
ay yee ng _ Ut 
(an 4) il ae 


* In another way I have deduced this relation in Ann. d. Phys. xvii. 
p- 496 (1905), formula (37). 


Theory of Surface Forces. 147 


We had: 


> “ 4 Ao oe a2 =) (see above). .. (24) 


Consequently : 


de—Tdn—HdS=3( p—p,)dv, + i(p—pi)dn + > fe od ( 


w+ = 


Up-— V1 


a Pia Cr Ue Vite Py Ps Mite, 
2 2 : Vg— TF Omg Vg— Uy nee. 
We have 
ee erent Pah) ay Pe Pees, CPP); 
and we get therefore: 
May dee ody Has 2 ey eae 
2 Vg— V1 
é Pi — Pe UAV, + Vd v5 Vy + (2p) U1 + V2 
+ Zz { Up Uy . 2 jue 
Further we have: 
V_Avy + U,V po + Vo ad + C2 20," dv, —2v,"*dvz aa 2v1V2 
Vg— 1} prone Vg—V, (vg—%)? Vg — Vy ; 


So the equation of energy of the capillary layer becomes: 


_— Dy) b a 
Tdy= =de+pdv—Hd8 + gota | ot tet 3 (a0) 


Ug— Vy Ug— VU} 


In the equation (28) the differential quotient sa is 


partial, the second parameter being constant. In the equa- 
tion (30) we must conceive R is the constant. The state of 
the plane capillary layer is completely determined by one 
parameter. This parameter is in the present consideration 
the temperature, and we may now state for the plane capillary 
layer : 


Tdyj=de+pdv-Hd8, . .. . (31) 


in this case : py=p2= ordinary vapour pressure. 
The equation (31) is also easily to be deduced directly. 
For if € is the thickness of the plane capillary layer and 


* The general equation of energy of the capillary layer becomes: 


Tdy=de-+pdv—Has—8(F3) ARBs g QW, | We, \ 


aR Y—t, V,-?, 


L 2 


148 Dr. G. Bakker on the 


p represents the ordinary vapour pressure, Spdé is the work 
done by p, and pxfd8 that done by pr. So we have: 
Tdn=de+ Spdt + prt dS. 
Now, however, 
= (p,.—pr)¢ and) e=er. 
So we get: 
Tdn=de+pdv—pldS+p, Cds. 
Further, for the plane capillary layer : 
Pi=Po=P = px= ordinary vapour pressure. 
Hence Tdyn=de+pdv—(py—pz)€d8, 
or Tdn=de+ pdv— Hds. > acct, 2) (31) 


For the curved capillary layer we may also deduce the 
equation of energy without having recourse to the equations 
(27) and (28). 

If as above §,, 8, and 8, are the total surface, which refer 
to the radii R,, R, and R, (per unit of mass of the capillary 
layer), we have for the work done by p, and py: 


S, p,(dR + sat) —8,p,(dR—4d6). 
For unchanged R consequently : 
2Sopoph + 3Sipidé. 
Further we have: 


CA an BE a0) 
S\= (1-5 + 7e)S and S= (1+) + 7p) 


The work done by p, and p, becomes consequently : 
: S 
ps (1 7 in) ae—3 (ri—ps) dé. 


For values of R of the order of a micron p;—pz is of the 
order y/o against p. For values of R of the order of a 
wave-length of light or larger, we may thus put for the 
work done by p, and p,: 

psdé. 

The work done by the pressure py (in a direction parallel 

‘to the surface of the capillary layer) becomes : 


2h h? me = ds, 2 J 
Best pcah (14 nt =) =ASiopo+ FS pr, (82) 


- "'g 
vo p= | pr dhe. 
1 


Theory of Surface Forces. 149 
2 
The members with - are again neglected. 


To find an expression for pr', we multiply the equation (2 a) 
with h and integrate with respect to h. In this way we find 


Rif po— Ri pr Ot S? po=pr'l, 
or as: R,=R—-32f, 
acs g c \- 
pi? =RE(1+ 52 )p2—RE( 1-35) 


By substitution in (32) the expression for the work done 
by pr becomes : 


Lg} aBy ( a) ds; 4 )p 
ASO prt = 1 oR pa— FRE FAG fan 
Neglecting the terms 


ie 
es 2R i 
with ip We have for the work done by 7p; : 


pat dS—(1— fp) (Px Ps) aS, 
and the total work becomes : 
pSde-+pdS—(1—f)6(P.—7s) dS. « (33) 
lf we had written the equation (2a) in the form: 


(R,—h) dpy=2( py—pz) dh, 


and had taken dh in the opposite sense to that adopted above, 
we should have found (by substitution of the expression for py’ 
in (32)) for the work done by pr: 


‘4 te No z, ‘ 
aS (1+ F) Spe SdS—a8Ep,( 1+ = 
The total work may be expressed thus by : 
pSde +pitdS—(14 2\e(p.—p)48, . (34) 


or if we take half the sum of (33) and (34) the work is 
expressed by : 

p(Sdf +£d8)—£(Ps—Px)dS*, . . (39) 

* For the plane capillary layer we have: v=s{ and H=(p,—p,)¢ 


The expression (35) becomes, therefore, for the plane capillary layer: 
pdv—Hads. | 


Further we have: = =j1— 


150 Dr. G. Bakker on the 


The equation of energy of the curved capillary layer 
becomes, therefore, for values of R of the order of a wave- 
length of light or greater : 


Tdn=de + p(Sdf + $d8)—P 5 RAS, . ee 


where R must be conceived as constant. 
Tf R is measurable or has a value not smaller than a micron, 
we have: 


Sd¢+¢dS=dv and PS R=u, 


and (36) becomes : 
Tdyn =de+ pdv— Ha. 
The equation (30), however, is contrary to (36) quite 
exact. 


§ 5. The plane capillary layer considered as the 
limit of a cylindrical. 

The formule simplify when we consider a cylindrical 
capillary layer instead of a spherical one. The condition for 
the equilibrium e. g. gives instead of the equation (11, § 2) 
the more simple one : 


Pi-po= (57) 
in the case when the liquid is inside the capillary layer 
(liquid drop). 


If, however, the cylindrical capillary turns its concave 
side to the vapour (bubble of vapour), we get : 


(p -¥ Pr) 4 
an ’ 


Pot, oP ty «i. ake 


In order to deduce the equation (37) we consider the 
cylindrical capillary layer simply like a tube under the 
influence of an inward pressure p,; andan outward one jm. 
Instead of p, and py, we put as above: p,and p,. If the 
corresponding quantities are represented by the same letters 
as in the consideration of the spherical layer, and z denote 
the total length of the cylindrical capillary layers of equal 
curvature, the condition for the equilibrium gives immediately: 


2 
2Ry2p; —2R,zp.= -={ pr dh, 2). ae (39) 
4 


or: —p 
Pri~ prs PEP a 
where R represents again half the sum Ri+R, of the inner 
and outer radii. 2 3 


Theory of Surface Forces. 151 
As for a cylindrical capillary layer the curvature is = and 


only the half of the curvature of a spherical awe layer 
with the same R, we have instead of the equation (2a)(§ 2): 


dpy (Ry +h)=—(pu-pr)dh. . . . (41) 


By integration with respect to h: 
9 2 % 
Ry(po—pi) + § po i} px dh= -{ py dh + { pr dh, 
1 


or: PA Pa= Ao ah ego. 37) 


In order to deduce the equation of energy we calculate 
firstly the work done by the pressure jy, p,, and pr. 

The capillary layer (per unit of mass) is determined by 
two parameters, and we consider the temperature to be 
variable while R=constant. 

The work done by pz and p, becomes : 


a8opodf + 48, pdf, 
or, whereas 8, = 8 (1+ ss] and S,=S (1- sn) 


pSdt+ , Sdt(p.—p). 


The work done by the pressure pr parallel to the capillary 
layer becomes further : 


e h Dat a 
ds, prdh (i+ = =ds iS prt oe pi? 


=a8(1— gh) te Sere * 


The total work becomes ne : 


psat— £ sat (p, — ps) +d8(1— 355) Set So pile’. (42) 


By multiplication of (41) by A and integration, we find 
easily : | 
R( po—ps) +30 prt3opo=tprotyprg, ~. (43) 
, 2 
vere yh ("pea 
Jl 


* For dS'=27R’dz, consequently 327R'dh prdz= 3 dS! prdh. 


L523 Dr. G. Bakker on the 


The capillary energy per unit of mass, or the volume 
integral of the departure py—pr from the law of Pascal is: 


2 2 h 
Hs= { S’ (px — pr) an=8, | (1 + x) (py—pr) dh 
1 1 1 


/ = peas 
a om Bey (Px—Pr) + 5p SF? PX’ — pr’), 


2 
ort H=£(Px—Ds)— gp {(Px— ie) —(os' =p}. (44) 
From (43) and (44) we easily find by elimination of py’: 


tp'= (R+M) p—(R-N)Pr— FH. . (45) 


The expression for the work done by the pressures p,, po, 
and pr becomes therefore : 


pSdf+sd8fp, + SdSE pr—3HdS + is SdZ (po—p) 


dS 
+82 tobe). ae 
Now we have: = pr =(R+40)p,—(R—1E) py. . (37) 
By substitution in (46) : 
p(Sdo+ CdS) —}HdS —} RdS( p,—p2) — a ( Pi—P2)t{28do + Eds 
(47). 
Further we have: 
v=n(R,?—R,’)z=7f (Ri + R.)e=2rlRz=LE; 
hence : dv=S d+ d8. 


The expression for the work done by the capillary layer 
becomes therefore: | 


R — 1 
feos. oS ae oe ad Pir ee ales ay me 
pdv pH(1 , Ras xf? apd (f0) *. 
The equation of energy of the cylindrical capillary layer 
per unit of mass becomes consequently : | 


Tdy=de + pdv—JH(1+ 9 )as—P\TPea (En), (48) 


* Por ¢ (28d(+(¢d8) = ¢ (dv+Sd{) = ¢dv+ rd(=d(vf). 


Theory of Surface Forces. 153 


The considerations, which the equations (27) and (28) 
produced, may be applied also to a cylindrical capillary layer. 
Quite in the same way as in § 4, we find again : 


Tdn=de+pdv—HdS + Bed { Zee tet, (30) 
(48) and (30) give : 
(R,—R} dS= ay =a oe . ee (fv). (49) 
On the other hand we have: 
De ee e50) 
Pi Pt Vas Gaye il 


where pxv denotes the volume integral of the pressure ps, 


or: 


This relation (50) we prove in the following way :— _ 

At the formation of the unit of mass of the capillary 
layer the quantities of liquid and vapour which have produced 
the matter of the capillary layer, were : 


pie Ie2 and.) et ee (see above § 4). 
P1—Pe2 Pi- P32 
The available energy which has disappeared is consequently: 


— { PPP P2 pe, ue . ae ub= bay eps i: + pe | 
PiyP2 


If p represents the average value of the saad of the 
capillary layer, we have for the last expression : 


5 \ 
ss + —— ome ay 
4p aoa Pa) " 


TE py = S'pr dh, we may consider — pv as the 


oS Ih 
available energ : which is gained at the formation of a unit 
of mass of the capillary layer. 

The change of the available energy at the formation of the 
capillary layer per unit of mass, t.¢. the capillary energy 
HS, consequently becomes : 


HS={ p+—2—(n.—pa)—pr be. s+ (52) 
Pi aps 


154 Dr. G. Bakker on the 
On the other hand we have: 


HS = (py — pr), 


and therefore : 


4 p | moh 
—_— — (: oe D5) . . . e ao 
meee ope 2 P2) (53) 


By means of the identity : 
P1 Pit Pe (CS" ow. ) 
+ ’ 
mt Pi—P2 2 
we find immediately : 
, . 
Pale pes, — 
MP2 Pi— Pr 
which is the equation (50). 
Now we have: 


v py= ’s Dx dh=S¢(1— aR) s+ 85 pv (54) 
Further we found 
R(p.—ps)— 38 Pu +430 po—t pr'S+ pr'l, . (43) 
and = H=£(ps—ps)— 25 { Bs —Pe—(pa'—px')}. (44) 
From (54), (43), (44), a (37) we easily find: 
Sea | 8’ pxdi=1H84 © io( 1 in) = ro Fa) 


(95) 
or by substitution of py in (50) : 


H—(p,—p,) )(R+ & iR =(p ps) Pee) (56) 


Further we have: =Rx. So the equation (56) 


becomes : Pre 


eta 2p — (pi + po) Li 
Be Re pa 


9 9 5 +2 
Peo are (Re-R)= Fo 4 cate yt | (58) 


Ug—— Vj Ug— Vy 


Theory of Surface Forces. T55 
By differentiation of (58) : 
Sd(R,—R)+ (Rx—R)dS=d ge +d) Bball, 31), Sr 


4R CoO Vg — Vy 


Now we had: 
(Re —R)dS=7p ace taf 20 pth (35) 


T.e., if the temperature is changed by constant value of R, 
we have: 


Sd(Rx—R) =SdR,=0. 


If R= Ha tlts remains unchanged so does also the Kelvin 
quantity Rx. At the critical temperature : 
x ma 
p= a 5) Pe and Pi—p2=9. 
Hence: mt) Pict Pe ee ge 
eal sien se 
Pia P2 d(p1— ps) 


In the neighbourhood of the critical temperature the 
. fa + Po « | Can =F 
difference p— i 9 Be is aluays small, and ales”) is 


therefore at the critical temperature null or finite. Further 
we have at the critical temperature : 


oF = ox and Pe = 
Hence 
Cte dahl 5— Pusha | 
lim a pel p 2 _ null or finite _ 0 
pra pi px! d(Pia Pai iasea ao” 
dT 


The equation (57) becomes, therefore, at the critical 
temperature : 


Cx? 
Rx hp dP fe P5 


where ¢, denotes the thickness of the capillary layer at the 
critical temperature. 


(60) 


156 On the Theory of Surface Forces. 


The equation (57) becomes thus : 


— 
J BRE ae 


If we now consider the plane capillary layer as a limit- 
case of the cylindrical of small curvature, we must take in 
(61) R=o. As (?—f always remains small, we have 

pple Pi 5 Pe 
lim. ————— =0 for R=. 
Pi— Pa 
So we have for the plane capillary layer p= Pee, 


I.e.: if we have a plane capillary layer the volume of 
which amounts to one cm**, this capillary layer contains as 
much matter as the totality of half a cm* of liquid and half 
a cm? of vapour. 

For the contribution of the liquid resp. vapour at the 
formation of the capillary layer (per unit of mass) (see 
§ 4) we found: 

a= PPPs and Lege Phe Pe 


Pi—P2 Pi— Pa 
For the plane capillary layer these quantities are thus: 


Pee 
(- =p being 5 ) 

Pi Bia P2 
Pit Pe Pit Ps 


Liquid and vapour contribute consequently to the plane 
capillary layer quantities resp. proportional to their densities. 

In consequence of the law of Cailletet and Mathias (the 
so-called law of the rectilinear diameter) p,+p: is in the 
case when the homogeneous phases of the liquid and the 
vapour are separated by a plane capillary layer a decreasing 
linear function of the temperature, or 


oats =pxte(Tx—T), 
where « represents a constant, px the critical density, and 
Tx the critical temperature. ; 


* For water at ordinary temperature the surface of the considered 
capillary layer becomes about 500 m’. . 


Bending of Electric Waves round a Large Sphere. 157 


We have thus for the plane capillary layer in the case 
when the law of Cailletet and Mathias applies: 
p=pxte(Tx—T), 

Or: 
The average density of a plane capillary layer is a decreasing 
linear function of the temperature. 


XIII. On the Bending of Electric Waves round a Large 
Sphere: Il. By J. W. Nicnouson, M.A., D.Sc.* 


Investigation of the transitional region. 


—T the section immediately preceding, the extent of the 
region of transition between brightness and shadow, 
when a radial oscillator is placed close to the surface of a 
perfectly conducting sphere, was examined. The present 
section is devoted to a discussion of the nature of this region. 
On reference to an earlier section this region, being the 
continuation on one side of that of brightness, will contain 


a magnetic force which, on this side, may be derived like (44) 
in the form 


yp = 3, u(lte"%) sin (mO— 4) e@™"%™"),—(G6) 
where 
u = iem(2m sin 6 RR, /wk?a*)?, 


provided that we neglect points in the immediate vicinity of 
the oscillator, so that 9 is not small, and the use of an 
asymptotic formula for the zonal harmonic is legitimate. 
A ditferent type of solution is valid for such points, which 
must be deferred for the present. 

Again, in the region of brightness, it was shown that the 
above series could be expressed as the sum of four others of 
the exponential type, such that two only could have a 
vanishing derivate of an exponent. This property will con- 
tinue to hold in the initial part of the transitional region. 
Since y, is not of an order capable of causing oscillation in 
an exponent involving it, the exponents of the four series 
may be regarded as dra—dnartm, whose derivates when x 
is not too nearly equal to unity are, with respect to m, or z2, 


sin7!vw— sin-ler+ 8. 


Only the lower sign of the ambiguity leads to a zero point 
at the boundary of the region of brightness. Accordingly, 


* Communicated by the Author; for Part I. see Phil. Mag. April 
1910, 


158 Dr. J. W. Nicholson on the Bending of 


the zero point in the continuation of that region is still 
determined from 
20=On—Gne— M0, . 2s 3) 
while it exists, and we may write, with this value 
y= S u(1 + e°!Xn) eh 
where on reduction, 
ee (AR, Ruy / 2 war? sin 0)? ‘ye OGD 


In this formula m=z has been inserted, for they now 
begin to differ by a quantity of lower order than z, and u is 
not oscillatory. The zero point, except in_ oscillating 
functions, is now sufficiently given by w=1. But x, can 
no longer be written as zero, for when #=1, it reaches the 
value 47. At the zero point it has a certain finite value yo 
not quite of order unity in this case. Again, under the same 
circumstances, 

v=Uyt+4huy (C—2)?, 
neglecting higher terms near the zero point a. 
The most significant part of the magnetic force becomes 


therefore 
oo 
y=u,(1+ e'Xo) zg el@ro | dE e2t2%0''® 


aaah 
= Up (1+ eX) (Qarz/ugl\ ota, | | (69) 


but its further examination is not important. It is a solution 
of the same type as in the region of brightness, but no longer 
represents the oscillator in the presence of a plane reflector. 

But on passing further into the transitional region, with 
6 still not small, d0¢,/d” and O¢nr/dn become of order m=, 
and can no longer balance @. The zero point therefore 
ceases to exist, and we enter upon the region more truly 
defined as that of transition in which we may expect, from 
ordinary considerations, to find bands of alternate maximum 
and minimum intensity, which, on the other side, merge 
into a continuous shadow. These bands will now be shown 
to occur, but before 0¢,/dn is of order m—s. 

In order that a set of harmonic terms in the series should 
become of supreme importance, it is not necessary that the 
derivate of an exponent should be zero, but-only that it be 
small. Hitherto it has not been possible for this to occur 
without the zero value being attained. But in the present 
case, we have just passed from a region in which the derivate 
can be zero to one in which it cannot. Although v’ cannot 
now become zero, it must attuin a certain minimum value e 


Electric Waves round a Large Sphere. 159 


for the component series which formerly had a zero point. 
On moving further across the transitional region towards 
the shadow, this minimum becomes larger, and the region of 
shadow may be designated as that for which this minimum 
is no longer a small quantity. This will appear subsequently. 
Meanwhile it may be shown that when ¢ is small, diffraction 
bauds must be present. 

The harmonic term whose order n is such that its exponent 
has the minimum derivative e will obviously be the “centre”’ 
of a cluster of important harmonics. The name “ minimum 
point” already suggested in an earlier section will be given 
to this, by analogy with the theory of the zero point. 
Denoting it by a suffix zero, it is evident that 


Vy =€, %/=0, v=Ute(a4—2%) +}09' "(a —2) 
in its vicinity, and the magnetic force will be derived from 
y= Da u(l+ e2txn el@ y 
where u has the value in (66), and 
20 = da—ar— mM. 


In the calculation of vp and up the expansions of the Bessel 
functions suitable for the case in which argument and order 
are nearly equal must be employed. These have been given 
by the writer* as follows :— 


TE fi = dw cos (w* + pw) | 


0 


A=\ dw sin (w* + pw) | Rte, Ch) 
sri 

ts =| dige ree 
Jo 


so that f, is an Airy’s integral f. 

Then if p= (m—z) (6/z)3, and m—z is not of order <, 
Im(z)=9-1(6/z)* A (p) \ ( 
J -m(Z)=2-"(6/z)3 {f; (p) cos mat (fo+fs) sin mat J’ 
and in the present case, since m=n+4 where n is an integer, 

J_n(2)=(—)'9 (6/2) (etfs. (72) 
the functions / having an argument p. 


* Phil. Mag. Aug. 1908. 
f Airy, Vamb. Phil. Trans. vi. p. 379; viii. p. 595. Stckes, Math. 
and Phys. Papers, ii. p. 329 et seg. 


71) 


160 Dr. J. W. Nicholson on the Bending of 
Again, 2sinmrK,,(z) e!""/r=J_»(2)—e "I m(<); 
leading on re iin m being half an odd aa a 
K,.(2)=2-? (6/2) (pt h—oi) ee". (73) 


Comparing with the usual substitution in terms of R, and 
o,, it follows that 


y= (2n)2 RE AEE) 
or= tan“" 7; /fs ' ie us 


where f, stands for fo+/;. If the functions F correspond to 
the functions f with argument pPi=(m—kr) (6/kr)4, then if 
2;=hkr 

1 


tan Pnr — F,/F,, } (75) 

Ruy = (2m)-2 684-3 (F+ FY). 

If an accent applied to any function denote differentiation 
with respect to its argument p or p, as the case may be, 


v' = 01/d2= (6/z)3 g!,—(6/21)3 bar — 0, 
since z—! 0p/dx=dp/dm=(6/z)3. 


Thus v=(; a wot A! | Gad 2) LO iad —F,F, 
Tae ft? +f? zy Fe+E2 
which may be reduced to a simpler form as follows :— 


The integral equivalent to w=f,—u/; has been shown by 
Stokes to satisfy 


—0, (76) 


u” — tpu= he, 
where the accent means 0/dp. Thus separating real and 
imaginary elements, 

i 1 —3pfi=0 


fe" — $e fo= —}. 
Again, it is not difficult to show that 
hs —3Pfs=3;, 
and by addition, since /;=/2+/3 
—3pfs=9, 
so that f, and f, satisfy the same equation, from which 
fh’ fifi! =9, 
and therefore J4/i’—/, fs’ is independent of p. 


Waves round a Large Sphere. 161 
But when p=0, 


=| dw cos w? = Sah.) 
2 0 2 


y 2 
ae =( wdw sinw = — aS r(5), 


ia ( dw (sin w? + e7") = s°(5) 


“0 
i= ( w dw (cos w+ e7") = = 5T(3), 
e/ 0 2 3) 


so that, by use of a property of the Gamma functions, 


Fit —fefl Hin =P PZB.) CD 


in a similar manner, and thus from (74-76) 
se eg te Wee kh FB) 


Otherwise, it is a known property of the asymptotic sub- 
stitution in R, and ¢, that 


0¢,/02=R;’, 
and since p=(m—<)(6/2)s 
roy 0¢,/0 2=0¢o,/Om= 6/2 2) 3 dn’ s) 


the accent denoting differentiation with respect to p, and 


0¢,/02= — (6/z)s (L—n— 2/32) hy’, 


and since the second term of this formula may be neglected 
if n—z is of lower order than z, even though p have the 
highest order consistent with this, it follows that 


aa 0¢o,/02= —Rz’, 


and the value of v’ in (78) follows at once. 

The comparison of these two modes of proof supplies an 
interesting indication of the degree of accuracy possessed by 
the formule (70 et seg.). It appears that even if p is of 
higher order than unity, provided that n—< is of any lower 
order than z, they may be used in the present case. 

This paper does not propose to tabulate the effect in the 
transitional region, for in the present state of our knowledge 
of the subject, the main interest attaches to a determination 
of the nature of the effect. The formule are therefore left 
in a somewhat undeveloped state for the present. 


Phil. Mag.S. 6. Vol. 20. No. 115. July 1910. M 


162 Dr. J. W. Nicholson on the Bending of 


Let ) denote the value of » at the minimum point, and 
Mp=N +4. 
Writing py=(m,—2)(6/z)3, pro = (ro — kr) (G/kr)s. (79) 
Then c—h, —k, —8@, 
Ry = (2m) 68a! f+ f) 
Ryr= (2ar)-1 63 (kr)—3 (F 24+ EY), 
zy,= tan“! f,/f,— tan-! Fy /F,— Om, 


v —_— =e 
fe) u tgp - i) 


y= as galt ( P R,Ryr/27a7? sin @)2, 
2 
1 j9 = Up eX 
tan yo= — 3 ORW/O2=(3/7)(AA thf), - + + (80) 


with the minimum point substituted in each. With these 
values, 


ye" a (w+ wy) eta 


a Tea) 
aay Le=, ail 
=7 anf ae i ger pute) ere ee 


by the usual summation formula, where the limits have been 
taken as infinite. Actually, they are not even of order z for 
the harmonics of the type contributing mainly to the sum, 
bat owing to the rapid oscillation of the exponential they 
may be regarded as infinite in the usual way. Again, to the 
same order, the multiplier of the exponential in the integrand 
may be taken as corresponding to the value of & making the 
exponent a minimum, or &=0. Thus 


La oe (Uy + Uyo)e" ie dg get iets Fe (81) 
and the integral is identical with | 


6 Ne CF ones : 
Bay) dw cos(w®+ow), . . .« (82) 
where o =z (6/zv9""')3, 


and y is therefore proportional to an Airy’s integral. The 
transitional region therefore exhibits maxima and minima 
after the manner customary in such problems when treated 
by the ordinary methods, provided that this integral is of 


Electric Waves round a Large Sphere. 163 


the type which oscillates. Now e is positive, and therefore 
for oscillation to take place it is sufficient that v/"’ should be 
negative. This condition is evidently satisfied, for v9’ has 
decreased from its positive value in the region of brightness 
to a zero value, so that its derivate is continuously negative. 
Otherwise, the result follows from (80). ‘The existence of 
maxima and minima is therefore demonstrated. But on 
passing further into the region, towards the geometrical 
shadow, e tends to increase, and the terms near the minimum 
point no longer have a preponderant sum. Moreover, 
x, tends to behave like g,. The oscillation between maximum 
and minimum ceases, and in fact the whole effect becomes 
of a smaller order of magnitude, the series hitherto neglected 
contributing to an equal extent. When e is of higher order 
than 273, the term of the exponential involving 1” 
becomes unimportant, and the sum is at once of lower order 
in so far as it depends on the series hitherto most important. 
It will appear later that the series tend to cancel one another 
in a remarkable way. 

The above investigation is restricted to points in the 
transitional region not too far away from the obstacle, for 
nr has been assumed to be of the same type as @, for a 
given value of n. Thus the bands will disappear ata sufficient 
distance. The investigation for the small values of @ is 
postponed, and the succeeding sections take up the problem 
of the geometrical shadow. 


Preliminary discussion of the geometrical shadow. 


In the geometrical shadow, as we have seen, it is not 
possible for the derivate of an exponent, in any of the series 
for the magnetic force, to vanish or become small. No 
‘group of terms, therefore, becomes of supreme importance 
after the manner of previous sections. Moreover, as was 
pointed out earlier, it follows that the harmonics of low order 
may contribute substantially to the sum, and thus the 
asymptotic expansion of the zonal harmonic must not be 
used even for a finite orientation from the transmitter. The 
proper formula was given in (48) and becomes 


yp= G (0) z m(RrRn-)? (1+ e2!Xn) sin mo o (dn—$dnr), 
7" 


where G (6) is an operation defined by 
2c sin? 0 do a — wdd (83) 
kar d@), /2/(cos8—cos¢)’” ° 4 
; M 2 


G (0) .w= 


164 Dr. J. W. Nicholson on the Bending of 


and y, does not lead to finite oscillation in the manner of 
¢. in an exponential. This series becomes 


yp=G (@) Xo 0 u Cee 


where 
u=m(R,,R,,-)2 (1+ e°tXn) /2e, 2 (Uy. Us) =hr— Pry mod. 


Since G(@) annihilates the first harmonic, n=0 is taken as 
the starting point instead of n=1. We restrict the investi- 
gation for the present to points actually on the surface of the 
sphere for this is the case to which controversy has hitherto 
related and which moreover has a much simpler analysis. 
Thus 

u=mR,, (1 + e7Xn) /20, 2(0), vo.) = Om ee 


Recalling the formula of summation *, which in the present 
case is very simple, since derivates of v higher than the first 
are all zero, the series 


S = 3u(e"@)... eee 


has a sum 


% 


S=: | de(U y+ Pay ti am (87) 


where €=Ny/Z, v=n/Z 


In the present case, n is to be replaced by n+ or m, so 
that 29 is replaced by 4. With the values of R, and yn 
developed in an earlier section when n is not comparable 
with <, 


ca ae 
) === r (1+ =), - . . Fy -° (88) 


where zv=n+4. More accurate values of the functions are 
found to introduce a correction which is only of order za’/z 
or x’, and cannot affect the argument of this or of the next 
section. There is no necessity to give the analysis of this 
point, which can readily be supplied from the higher approxi- 
mations already given. It will be necessary later to use 


* Messenger of Math. Oct. 1907. 


Electric Waves round a Large Sphere. 165 


these higher approximations in a more extended investiga- 
tion of the region of brightness. 

Before proceeding further at this point, it is necessary to 
make some remarks about the sum of the series. Althou oh 
there is no group of terms of supreme importance on account 
of a vanishing derivate of the function v, it might be thought 
that since wu rises in order in the neighbourhood of any pole 


which it may have, terms in such a region might supply the 
important part of the sum. But the poles of uw must arise 


from the zeros of the function 
Hide a2 Wk (2), 


none of which are real. Moreover, their imaginary parts 
are all large, so that such terms will necessarily have an 
exponential factor in their sum, of large real argument. 
This argument cannot be positive ‘from elementary phy sical 
considerations, and therefore it is negative, and the sum of 
any such set of terms decreases in a rapid exponential manner 
round the surface, and may be ignored in comparison with 
the terms considered in this section. The rigorous analysis 
of this point will follow later, these rough indications being 
enough for the present. 

Since in the integral of (87) there is no zero point, and 
no difficulty introduced by poles of the function wu, an inte- 
gration by parts is legitimate. This is taken between the 
limits e and infinity, corresponding to n=0 and n=, and 
e is in this case (2z)-1. The term at infinity may be 
neglected as corresponding to harmonics of infinite order, 
which have been dealt with already, and for the term at 
v=1/2z, the value of wu in (88) may be used. 

The order of the integral thus becomes that of 


Deer da: or i ; 


€ wee 
where 20. = ip, and Us = a= (22) y, 


rejecting 2~* throughout. The integral thus has zero order 
in z at most, and by an inspection of ‘the operation G(@), yp 
would therefore not be of greater order in linear magnitudes 
than (ka?)—!. When the sphere is absent, the corresponding 
order of this quantity is known to be &, as found in the 
investigation of the region of brightness. Accordingly, the 
magnetic force when the sphere is present is at most of 
order (ka)~? relatively to its value when the sphere is absent, 
and this is approximately 10- in the numerical case typical 
of wireless telegraphy. 


166 Dr. J. W. Nicholson on the Bending of 


The shadow produced by the sphere is therefore very 
complete, and in fact much too complete to admit diffraction 
as an explanation of the experimental results. For very 
small orientations, of course, the exponential portions of the 
sum could be of greater order than this, and they are not 
included in this remark. 

The integration by parts remains valid for harmonies in 
the vicinity of m=, although R,, and therefore w, is of 
higher order in this case. Tor the change in R, is con- 
tinuous, and it is only necessary to break up the integration 
from € to « into several stages in which different formulze 
for R, are used. The continuity of the values of R, and 
o, is demonstrated in a paper on the asymptotic expansions 
of Bessel functions *. Since the term neglected in the inte- 
gration is necessarily of the same order relatively to that 
retained, in each part of the range, a series of integrations 
by parts is sufficient to show that “the terms near m =< are 
already fully taken into account. Their main effect is in 
fact exponential, as will appear later. This investigation is 
sufficient to give an upper limit to the value of the diffracted 
effect, and is complete enough to decide the main point of 
controversy on this subject. It will be noticed that even 
when points not on the surface are treated, and $,— qn» is 
therefore not zero, the effect cannot be of a greater order 
than above, so that the use of a receiver not very close to the 
surface cannot, with diffraction alone as an aid, furnish 
an effect sufficiently great to be perceptible, for finite 
orientations. 

But for other purposes, it is necessary te carry the calcu- 
lation further, and to obtain an actual formula for the effect, 
and it will be shown in the next section that the actual effect 
is much smaller than the limit assigned above. 


Further examination. 


We proceed to collect the terms of the order which is 
apparently most significant in the expression fcr the magnetic 


force. Since by (88) 
| 24 ie 
~ feat 
neglecting «° and zx°, therefore 
te (ee 
Du, = (e/t) (1+ ga") = 2/t, 
Pra == a) Abe 


* Phil. Mag. Feb, 1910: 


Electric Waves round a Large Sphere. 167 


and on examination, no further derivate is of higher order 
than z. Thus to order zero in z, from (87) 


s eo 


Uot+ +... Suny + Du, . Wit Du, . ub +... 


ee This becomes, in terms of v’, on 


reduction, , erininih: 
(v' cos $v’ —2 sin $v')/4 sin’? gv’. e72”, 


and therefore from (84, 85), if (v), %)=+ 6, 
yp = G(@) ef” diz ee pid ..)(e —¢#"2) 
U1 errr ne 20 
= (0) [(Uo+ = =f ey MS wry! i =| 
ae 
= G(6). @ COs 16- 2sin4 (2 + +1 Jao, 


At sin? 5 
so that the term of this order in the magnetic force vanishes. 
Moreover, it may be shown at once that whenever 


is the product of e~*” and an odd function of v', the same 
result must occur. A further examination of the magneti 

force indicates in this way that the set of terms of order <7! 
contributes zero when v'’=0, or when the point at which the 
effect is desired is on the surface of the sphere, so that 
$,=¢,» The magnetic force is thus at least two orders 
higher than was shown in the last section. 

These results indicate that the vanishing of these terms of 
successive orders is general, and this will now be shown to 
be the case, by an independent method. | 

Consider w as a function of m or n+4. It may be shown * 
that if R, is derived from Bessel functions of real order m, 
without any restriction of m to half integral values, 


R, = -- ( “Ko(2zsinht) cosh 2mitdt. . « , (89) 
“0 


This integral is an even function of m, and so also therefore 
is R,. Moreover, x, is also an even function because 


tan Xn = at TOR,/0<. 
Thus w, which is proportional to mR,(1+ 7a), is an odd 
* Phil. Mag. Feb. 1910, p. 234. 


168 Dr. J. W. Nicholson on the Bending of 
function of m, and may be expanded in an asymptotic series 
in the form 
mAg+mA,+m?Ast+.. 
so that 
ye = G(@) Be (mAgt+mA,+...) sin md, 


where the coefficients A are independent of m. 
Now it is known that, when convergent, the series 


8 


> m?*! sin n_ree yO. e 


m=V 


- (90) 


if p is an integer. 

Now near the lower limit e¢ of the integration in the 
previous section, it is known that the coefficients A decrease 
in order of z, from the form of R, asa function of m when m 
is small. It follows that the terms of yp expressed as a series 
in descending powers of < will continue to vanish, and this 
shows that the actual value of yp must be of an exponential 
form for points on the surface of the sphere. Thus the result 


of M. Poincaré’s revised iny estigation, and that given by the. 


method of this paper, are not at variance for surface points, 
and the effect in this special case is entirely exponential. 

But this line of argument is liable to failure when the 
magnetic series contains an oscillating exponential of argu- 
ment proportional to ¢,—@¢,,, and it is therefore not yet 
shown that the effect at other points in the shadow is deter- 
mined by an exponential law. M. Poincaré’s mode of proof also 
fails, in its present form, for other than surface points, as it 
definitely assumes the absence of oscillation. 

In the next section, a preliminary discussion of the 
exponential sum is given, and the investigation for other than 
surface points is postponed for the present. 


Determination of the exponential sum. 


As the sum of the harmonic series, for points on the surface, 
is now shown to be mainly caused by terms in the neighbour- 
hood of singularities of the function uw, it is simpler to proceed 
otherwise at this point, as a direct summation may be effected. 
Let v be a typical value of m making 


djdz. 2K,(2) 208 «oo 


Then m=v is a pole of the function uw, and this must be a 
simple pole because zw is proportional to the ratio of -*K,,(z) 
to its derivate. Again, on reference to the original expansions 


Electric Wares round a Large Sphere. 169 


in R, and @, for the Bessel functions, it is seen that the poles 
are given by the sclution of the equation in m, 


Bi 2 == re €72 


and since, whether m be real or not *, 
R, Lt ae | ‘ 
KG sinh ¢) cosh 2m dt, 


we note that R, is an even function, so that m=-—y is 
also a pole. The poles thus occur in pairs. Moreover, an 
inspection of (89) shows at once that it cannot be satisfied 
for real values of m when z is real. Thus there are no real 
poles. Now when m, even though complex, is not nearly equai 
in modulus to z, although less, 


R, = 2/(2—m?*)= 


and it is found to be impossible to satisfy (92) within the 
limits of validity of this formula. The poles therefore 
correspond to values of m of order z at least. 

We may assume, at once, the justification appearing in the 
result, that the poles contributing mainly to the sum are 
those whose i imaginary parts are least. Mor eover, it is fairly 
evident from the above reasoning that the least order the 
imaginary part can have is that of 2, since there are no poles 
in the first region of expansions of the Bessel functions. In 
the section followin @, the first pole is determined and found 
to be of the form 7 


i = 2— tener a SO (GS) 


where fo is a numerical quantity approximately equal to 1/3, 
and its contribution to the sum is of supreme importance. 
We shall also, for the present, assume that the imaginary 
part of the poles is negative. If a, be the residue of the 
function 


4 1) cee Dene 
AK (2) /S «AK, (2) = HR. + 20) 


at the pole v, then the corresponding terms in its development 


by Cauchy’s theorem, including the pole —yv, become 


1 1 2va, 
Cy Tis we ee oe 
mV m+y m*— y- 


* Vide Phil. Mag. Feb. 1910 for the case of m real. The proof there 
given can obviously be extended. 


170 Dr. J. W. Nicholson on the Bending of 


Thus again defining G(@) by the operation 


: eesin @ d {7 wd 9, 
G(O).w = ag al Vv 2./(cos @—cos¢)’ (94) 


Then for points on the surface, by (83), 


yp = G(@) S mR, (1 + e?'Xn) sin mg 
u 


r= 
ae 2G(6) Ss 2va, 4 
= — 21G(6) 2 2y ae min md, 
ru= 


and as msin mo/m?—v? is an even function of m, the 

summation may be replaced by half that from —2 to o. 
Therefore 

4 PO! a) STE 2 

fp -1G(0) >, 2va, Si = e, ee) 


~ 2 
m=—o MM —V 


where 2m takes all possible integral values, positive or 
negative. But the last summation may be effected. For 
consider the function /(v) sec xv, where /(v) has no poles. 
The poles of the function are given by cos@v=0 or 


my =+ (n+4)r = + mm, 


where mm is a typical value of m in the desired summation. 
The residues at these poles are /(-+m)/2 sin mz, so that 


sas a ] f(m H—m 
TOONS OT a ae a ak ee (2 ) ff ie ), 
: _ wsin mar\v—m v+m 
m taking half integral values. Identifying this with 
the series 


o =F; 20) 
< msin in i uf = ( 
Poet Q<— 
a we ~_ 


snmp sin ind 
(ee) 


m? —y" wo \ Vm v—m 
we find at once that 
2 m sin m 7 cos v(741—d) 
= bes g => i. p 5) e . . . (96) 
eae — COS VI 


and therefore 
yp = —2mtX,va,G (9) . sec vr cos v(7—¢). 
Assuming the result of the next section, that the first pole 
is one for which v= z—1:38, where # is an ordinary 
numerical quantity, we have, @ being less than 7 throughout 


Cy 


the operation of G(@), 


Electric Waves round a Large Sphere. 171 


and is very small, containing an exponential of large real 
negative argument. Thus 


yp = —2mtd,va,G(O)e—'"? 


oO 8 6 a4 al 
ka db” ” J, 4/24/(cos 0—cos b)’ 
and the important part of the integral is contributed near the 


lower limit. Writing 6=@+€ and neglecting square and 
higher powers, the integral becomes 


: ae A a pe TN st) = aah Ae : eat 
Je s4(26sin 0) ° TD : 


and taking the leading term in the differentiation with 
respect to oo e finally obtain 


—2sin 6 Me ee 1 
0 = a Oman ot pean 2, .. 197) 


and it is now necessary to determine the residue a,, 

It is already obvious from the last equation that the poles 
whose imaginary part has an order greater than zs are not 
important, and they will henceforward be neglected, and the 
summation restricted to those poles which are of type 


a Za 8, 


8 being of zero order and numerical. 
Now near m=v, by definition of a,, 


a, 


Wey 


a ae 
2Kn(2)/ $2 -8Ku(2) = 


and therefore 


== 22 Kmn(2 525: 2K,,(z) 


with m=v substituted after the differentiation. But v being 
of the above form, the Bessel function is proportional to, by 
results referred to earlier, 


2~#f(p), 


where p = (m—z)(6/z)s, and f(9) does not otherwise contain 
mor z. Thus 0/dm = (6/z):0/d¢e, and moreover, so far as 
the term of highest order is concerned, 0/0z= ~(6/z)3 ae 
For in this term, differentiation only lowers the order by ¢ 


172. Bending of Electric Waves round a Large Sphere. 
and in others, by -<. Thus we may write near m=y, 


0/02 = —0d/dom, 


and accordingly, 


by the use of the differential equation satisfied by the 
function K. This result is very elegant. To the same order, 
we may write 
; weft ~2 2) cet melaye a) 
a, = 2/(2—v") = FB We ie 


on reduction, and therefore 
y = k2(ka) —i(Q7 sin 0) TB vie WO + ae, 


But in non-oscillating terms, we may write v=ha to the 
order already retained, so that finally 


y= hk? (ka) —% (Qa sin Q) 1 ¥, Boe Fo 80 — had + jer (99) 


and only the first term of the summation is really important. 
For an undisturbed oscillator, the corresponding formula 
becomes 


Yo = —1k2(ka) cot 1Qe— Pha sin 20+ 2e7 ; (100) 
and the ratio of the amplitudes in the two cases is therefore 
(87 sin 8)2(ka)i tan 44 . Be Aa380 (101) 


and the exponential factor is of the same form as that derived 
otherwise by M. Poincaré, who does not give the other 
factors nor the value of 8. The impossibility of explaining 
the experimental results by means of diffraction is now 
evident. In the next section, a determination of 8 is made, 
and an examination of the formula numerically is given. 
Succeeding sections deal with the remaining problems 
hitherto postponed, viz., the effect at any point in the 
geometrical shadow, the effect in the neighbourhood of the 
oscillator, and the determination of a second approximation 
for points in the region of brightness. 


pales | 


XIV. Vhe Heat developed during the Absorption of Electrons 
by Platinum. By O. W. Ricuarpson, M.A., D.Se., 
Professor of Physics, and H. L. Cooxn, M.A., Assistant 
Professor of Physics, Princeton University ™. 

[Plate III. ] 


§ 1. ie 1901 + one of the writers showed that the pheno- 

mena attending the emission of negative electricity 
by hot metals could be explained on the assumption that the 
electrons which, on the electron theory of metallic conduction, 
move freely inside the metal, attain sufficient kinetic energy 
at high temperatures to enable them to overcome the forces 
tending to keep them inside the metal, and so escape. From 
the way in which the thermionic current varied with the 
temperature it was shown that the difference, w, in the value 
of the potential energy when outside and when inside a metal 
could be calculated. Somewhat later t it was shown that the 
existence of this difference in the potential energy would 
involve a loss of thermal energy by the substance when the 
electrons were being given off, and it was pointed out that 
this effect would increase very rapidly with the temperature ; 
so that at sufficiently high temperatures the loss of energy 
due to this cause would be greater than that arising from 
thermal emission. An effect of this character has recently 
been discovered by Wehnelt and Jentzsch §. 

Another consequence of the existence of this difference of 
potential energy is that when electrons possessing negligible 
kinetic energy pass into a metal an amount of heat should be 
developed which is equal in magnitude to the difference in 
potential energy for each electron multiplied by the number 
of electrons entering the metal. The present experiments 
show that this effect exists, and is of the expected order of 
magnitude. 

On this view of thermionic emission, the loss of energy 
when the electrons escape will consist of two parts :— 
(1), that due to the thermal kinetic energy of agitation of the 
escaping electrons, and (2), that required to overcome the 
work function, w- Recent experiments show that the former 
is equal to the kinetic energy of thermal agitation of a 
molecule of gas at the temperature of the metal, and corre- 
sponds, at any temperature which is available experimentally, 
to the energy which would be acquired by falling through a 

* Communicated by the Authors. 
y+ O. W. Richardson, Camb. Phil. Proc. vol. xi. p. 286. 


t O. W. Richardson, Phil. Trans. A. vol. cci. p. 497. 
§ Ann. der Physik |3] vol. xxviii. p. 537. 


174 Profs. Richardson and Cooke on the Heat developed 


potential difference of a fraction of a volt. The precise value 
of the fraction depends, of course, on the absolute temperature, 
to which it is proportional. The first loss appears to be small 
compared with the second at all available temperatures. 

The heating effect which should occur, on this view, when 
electrons enter a metal is made up similarly of two parts : 
one proportional to the kinetic energy, /, which the electrons 
possess immediately before entering the metal, and two, a 
part which depends upon the difference of potential energy 
of the electron when inside and outside of a metal. This will 
be equal to the work, w=ed¢, done by the surface forces on 
the entering electrons. Here e is the charge on an electron 
and @ is the difference of potential energy per unit charge. 
Hence, if » electrons enter the metal in unit time, the rate 
at which kinetic energy is transported into the metal will be 
equal to n(epb+h). In general, & will consist of two parts, 
the first due to the energy, «@, of thermal agitation of the 
hot metal from which they are emitted, and the second, due 
to the potential difference, V, driving them from the hot 
metal to the cold. If the conditions are steady, the current 
will be continuous and there will be a flow of an equal number 
of electrons out of the cold metal, each of which will trans- 
port on the average an amount of kinetic energy «6 with it, 
where 4 is the temperature of the cold metal. The total 
quantity of heat developed by the n electrons when they flow 
into the cold metal is, therefore, equal to 


2? (ed + eV+ a( 0; = 9) ), 


so that the rate of production of heat by a current, 2, will be 
equal to . 
i(o+V4 : (8-0) ). 


For constant values of 6, and @) the rate of production of 
heat will thus be a linear function of V which takes the value 


@ + : (0,—9), when V = 0. 


This development of heat is clearly analogous to the heat 
liberated during the condensation of a vapour to a liquid. 


§ 2. Method of Haperimenting. 


To detect and measure this heating effect an electrical 
method was used. The electrons were obtained by heating 
osmium filaments which were kindly presented to us by the 
Deutsche Gasglihlicht Aktiengesellschaft of Berlin, to whom 


jon 7 


during the Absorption of Electrons by Platinum. 175 


we are glad to be able to take this opportunity of expressing 


our thank Osmium is a very refractory substance, and 


when a filament of it is heated if emits a copious supply of 
negative electrons which can very easily be regulated by 
adjusting the heating current through it. A long narrow 
strip of thin platinum: foil was wound in the form of a orid 
on glass supports so that the different strands were nearly in 
one “plane. Two osmium filaments (see below) were suitably 
supported, one on each side of the grid, and insulated from it. 
The filaments were heated by means of an electric current, 
and an adjustable difference of potential could be applied 
between the filaments and the grid, causing the electrons 
emitted by the filament to flow into the orid. The ensuing 
rise in temperature of the platinum grid was determined by 
measuring its resistance by a Wheatstone’s bridge method. 
The present arrangement differs from the usual Wheatstone’s 
bridge problem, however, in one important respect. In 
addition to the usual battery current flowing through the 
four arms, we have also the thermionic current, which flows 
from the filament into the strip which forms the other arm 
Unless the effect of this is compensated, there will be a 
deflexion of the galvanometer, even when the resistances are 
in the proper proportion for a balance to be obtained under 
the usual conditions. This compensation was effected by 
introducing two auxiliary resistances into the bridge in the 
manner described below. 

In addition to measuring the change in the resistance of 
the strip produced by the electrons, it was also necessary to 
measure the thermionic current which produced it. This 
was done by means of a micro-ammeter suitably inserted 
between the bridge system and the positive terminal of the 
cells which were used to maintain the applied difference of 
potential between the heated tilaments and the strip. 

The rest of the arrangement will be made clear by referring 
to the accompanying diagram (fig. 1, p. 176). The electric 
current which heated the osmium filament, F, was supplied 
by the battery B, and could be regulated by means of the 
rheostat J. The voltmeter V, served to measure the potential 
drop along the filament. The strip, indicated by 8, forms one 
arm oi the Wheatstone’s bridge of which R,R,R; are the other 
arms. G is the galvanometer, and D, C,, C, the battery arm, 
R,and R; are the compensating resistances which are respec- 
tively connected to the ends of the galvanometer arms .The 
junction between R, and R; is connected through a switch 
to the mi¢ro-ammeter N, ‘which serves to measure the 
thermionic current. The other terminal of this was connected 


176 Profs. Richardson and Cooke on the Heat developed 


to the positive end of the battery B,, which supplied the 
potential difference necessary to drive the electrons from the 


Hie. 1, 


filament F to the strip 8. B, was in series with the rheostat 
H and the potential could be tapped off from various points 
of this by means of a sliding contact. The switch A enabled 
the sliding contact to be connected with the negative end of 
the filament F and the potential thus applied between F and 
S was ineasured by means of the voltmeter V,. The rocking 
switch A was introduced so that this potential difference 
could be changed from zero to any desired value almost 
instantaneously, and thus the flow of the electrons into the 
grid could be started or stopped when desired. 

The mechanism of the compensating device needs a little 
further consideration. Imagining for a moment the therm- 
ionic current from F to § to be turned off, it will be seen 
that the condition that the battery O;, C, should produce no 
deflexion in the galvanometer G is the usual relation 
S: kh, = R,: R,. Now sappose the thermionic current to 
be turned on. There will be a current flowing into various 
parts of the resistance 8. The result will be that in general 
the two points K and L will no longer be at the same potential. 


during the Absorption of Electrons by Platinum. 117 


There will be a point M in 8S such that at this point the 
current along S which arises from the thermions is zero. The 
thermionic current will flow from M towards R,, and towards 
L, and from those points along the various possible connexions 
to the point, wherever it may be, (in the present case at FE) 
where the thermionic current is led out of the circuit. It 
will be seen that if the resistances R, and R; are introduced 
as shown, provided the resistance trom M through R, to K 
is to the resistance from M to Las R, is to R;, K and L will 
still be at the same potential even when the thermionic 
current is flowing. 

The method thus involves a double adjustment which was 
carried on as follows :—First of all the switch A was put to 
the left, so that the thermionic current was off. The resistance 
R, was then adjusted until no current with the battery C,C, 
on flowed through the galvanometer G, involving the usual 
condition of proportionality between the arms 8, R,, R., and 
R;. The battery C,C, was then put out of commission, and 
one of the resistances Ry, R; was adjusted until no deflexion 
of the galvanometer occurred when the thermionic current 
was turned on. It might be thought that if the battery circuit 
DC, was broken when the second adjustment was made, the 
adjustment would not hold when the switch D was subse- 
quently closed, owing to the resistance in this arm being 
different under the two sets of circumstances. To avoid this 
objection two separate batteries C,, C. were used, and it was 
arranged by means of a commutating switch that they could 
either be put in series or opposed to one another. In the 
one case the potential in this circuit weuld thus be twice that 
of a single battery while in the other case it would be zero, 
but in either case the resistance of the arm DC,C, would be 
the same. In carrying out the first adjustment the batteries 
C, C, were in series and produced the bridge current, while in 
carrying out the second adjustment they were opposed to one 
another, so that there was no current actuating the bridge 
circuit. As a matter of fact, we were not able to satisfy our- 
selves that it made any difference whether the switch ID was 
open or closed when the second adjustment was made. 

In order to be quite certain that this method of com- 
pensating for the effect of the thermionic current flowing 
into the arm 8 was free from objection, we made dummy 
experiments with a resistance in the arm 8 provided with a 
sliding contact, so that small differences of potential from a 
battery could be introduced between the contact and the 
point E. The operations were then repeated with this 
arrangement. First the resistance R, was adjusted with the 


Phil. Mag. S. 6. Vol. 20. Ne. 115. July 1910. N 


178 Profs. Richardson and Cooke on the Heat developed 


cells C,, Cy on, so‘that no current flowed through the 
galvanometer G. The cells ©,, C. were then made to 
oppose one another, and it was found, as was to be expected, 
that when the battery between E and the sliding contact was 
turned on there was a deflexion of the galvanometer G which 
could be stopped by suitably adjusting R,. It was now found 
that after putting C, and C, in series there was no deflexion 
in the galvanometer G, whether the potential difference 
between E and the sliding contact was on or off, showing 
that the method of compensating worked satisfactorily. 


EOC. DOE 

G G 
The construction of the part of the apparatus containing 
the grid and filaments is shown in detail in figure 2. The 


during the Absorption of Electrons by Platinum. 179 


whole was in a cylindrical brass box B connected through a 
side tube with a McLeod gauge and Gaede pump. The 
supports of the grid and filament were fixed air- tight with 
sealing-wax through four tubes in a glass stopper A which 
was eround to fit a conical hole in the top of the brass box. 
This was found to make a perfectly satisfactory air-tight 
joint if the two surfaces were carefully ground and suitably 
greased. The glass tubes DD supported a glass framework 
on which the strip S was wound in the manner shown. By 
twisting the glass rods which formed the framework very 
quickly while they were being drawn out it was found 
possible to produce a thread on them in which the strip could 
be laid, and which prevented the possibility of its slipping 
about after it had been wound. The return end T of the 
strip was prevented from touching the intermediate portions 
by being wound around two glass projections as shown. The 
ends of the strip were soldered onto two platinum wires 

* sealed through the tubes, which were filled with mercury, and 
thus made contact with the outside. 

The osmium filaments F were soldered onto the outside of 
two bent brass tubes which were clamped together by two 
glass plates K, bolted at L. The inner side of the brass tube 
was cut and opened out as shown in section below. ‘The 
outer tubes were made so as to just slide on the brass rods C. 
This arrangement carrying the filaments was placed around 
the closed framework carrying the strip and slid onto the 
brass rods C, being fixed in position by the screws H. It 
was adjusted so that the two filaments F were opposite the 
middle of the grid S. The object of the inner tubes G was 
to shield the glass supports so that the electrons did not flow 
onto them. It was thought that if this happened some of 
the heating effect might be lost. 


§ 3. Method of taking Observations. 


In taking the observations it was found necessary first of 
all to wait for a considerable len eth of time for the tempera- 
ture of the system to become steady. This was usually a 
matter of two or three hours after the heating current 
had been turned on. Up to that time the resistance of 
the strip S gradually increased. In fact, it was generally 
found that no matter how long one waited there was a slow 
drift in the direction of increasing temperature and in any 
case there were apt to be small slow alterations one way 
or the other. After the lapse of an hour or two, however, 
they would be so small as to be of no serious consequence 


N 2 


180 Profs. Richardson ond Cooke on the Heat developed 


with the method of taking galvanometer readings described 
below. When the conditions had become satisfactory the 
resistance Ry was adjusted first of all, so that there was no 
deflexion of the galvanometer when the Wheatstone’s bridge 
E.M.F. was operative. Then the cell C, was reversed and the 
balancing resistance R, adjusted until there was no deflexion 
of the galvanometer. The battery C, was now reversed 
so that C, and C, were in series, and since the direct effect 
of the thermionic current on the galvanometer has been com- 
pensated for, any deflexion produced when the thermionic 
current is turned on can only arise from the alteration its 
heating effect produces in the resistance of the stripS. There 
will, of course, be some heating eftect due to the increase of 
the current flowing through the strip on account of its Joule 
effect. It is easy to show that with the current used in the 
previous experiments this effect was small compared with the 
effects measured, and the correction for this will be considered 
later. On turning on the thermionic current it was found 
that the balance of the galvanometer was disturbed, the 
deflexion being very rapid at first but gradually dying away 
to a small drift which was difficult to distinguish from the 
gradual drift of the galvanometer. Preliminary experiments 
showed that much the greater part of the heating effect 
occurred in the first half minute. It is difficult to be certain 
exactly how much, but, with a thin strip, 90 per cent., and 
probably more, of the total change of resistance developed took 
place within this time. Presumably, in difterent experiments 
under similar conditions the same fraction of the final rise of 
temperature would be developed in equal times, so that the 
method was adopted of always measuring the increase of 
temperature, or the change of resistance, to which it is pro- 
portional, developed in a given interval. This was usually 
30 seconds, but in some of the earlier experiments only 
15 seconds, while more recently intervals of one or two 
minutes have been used. As a matter of experience, how- 
ever, it seems that 30 seconds is the best interval to use as it 
is long enough to obtain the bulk of the effect, and to enable 
the periodic oscillations of the galvanometer to die down, 
while avoiding to a very considerable extent the errors arising 
from the drifting of the temperature to which the heatings 
with longer intervals are more subject. We have made 
several experiments to test the point, but have not been able 
to convince ourselves that there is any difference in the values 
obtained when different times are employed, except such as 
might arise from casual fluctuations. In order to eliminate 
the effect of these as far as possible, long series of readings 


during the Absorption of Electrons by Platinum. 181 


were taken, first with the thermionic current on for the given 
interval, then with the thermionic current off for the same 
interval, and so on in succession. In this manner the effect 
of the drift and casual fluctuations could be eliminated. 

In the first experiments the heating effect of the thermionic 
current was compared with the heating effect arising from 
an increase in the current produced by the batteries C,, Cy 
of the Wheatstone’s bridge. This current could be increased 
or diminished at will by altering the resistance in the box R 
in the bridge arm of the circuit. The heating effect pro- 
duced in this way in the strip 8 is equal, of course, to the 
resistance of the strip multiplied by the difference of the 
squares of the current passing through it, and could thus be 
calculated, the currents through the bridge being measured 
by a suitable milliammeter. It was found, however, that the 
change of resistance thus produced did not vary with the 
time in quite the same way as that due to the thermionic 
current, but got up to its maximum value somewhat less 
rapidly. There is an important difference in the mode of 
liberation of heat in the two cases. That due to the ther- 
mionic current is developed at the surface of the metal, 
whereas the other is a volume effect. On these grounds we 
should expect the final state of equilibrium to be reached in 
different times in the two cases, so that there is an objection 
to the measurement of the heating effect by comparison of 
the effects produced in equal times in the two cases. 

There is, however, a simple method of deducing the heating 
effect arising from the difference of potential energy of the 
electrons inside and outside of the metal, which appears to be 
free from this objection. We have seen that if a difference 
of potential is applied so as to drive the electrons from the 
filament F to the strip §, their kinetic energy will be in- 
creased by a calculable amount, which is proportional to this 
difference of potential. So that if we compare the effect 
produced by the electrons when they fall through no voltage 
with that produced when they fall through a voltage V, we 
shall at once be able to determine the difference in the 
potential energy in terms of the kinetic energy gained by an 
electron when it falls through a potential difference of one 
volt. When the difference of potential driving the electrons 
is zero, the thermionic current is so small that it is impossible 
to measure the heating effect to which it gives rise with 
accuracy. But this difficulty can be avoided if we compare 
the heating effect produced by the thermionic current with 
two different voltages. On the view developed at the be- 
ginning of this paper, the effect per unit current will be a 


182 Profs. Richardson and Cooke on the Heat developed 


linear function of the voltage, and from the position of the 
point at which the heating effect produced by unit thermionic 
current cuts the voltage axis, on the diagram in which this 
effect is plotted against the voltage, we can at once deduce 
the difference of potential energy in terms of the work done 
on the electron when it falls through one volt. All that is 
necessary, then, is to measure simultaneously both the ther- 
mionic current and the change it produces in the resistance 
of the strip S for a series of different voltages. This method 
has the advantage that in every case the heating effect is 
produced at the surface of the strip, so that the conditions 
are more comparable than when the heating effect of the 
thermionic current is compared with the heat ‘production due 
to an increase of current in the Wheatstone’s bridge. 

It is also simpler to work with, because the sensitiveness 
of the galvanometer depends on the current actuating the 
Wheatstone’s bridge; so that in measuring the heating 
effect of an increase in this current, it is necessary to deter- 
mine the sensitiveness of the bridge for each current used, 
The experiments made with this method of standardizing the 
effect agreed as to order of magnitude with those which 
depend on a direct comparison of the effects of the different 
voltages. There was, however, a definite difference in the 
magnitude given by the two methods, which we believe to be 
due to the fact that the mode of liberation of the heat is 
different in the two cases. 

It was stated above that when there is no voltage driving 
the thermionic current the effect is too small to measure 
with accuracy. It can, however, be detected and measured, 
and it is found to agree with the value determined by the 
less direct experiments within the order of accuracy of its. 
measurement. 

Another point which was tested in the preliminary ex- 
periments was whether the change of resistance in the strip: 
produced in a given time was proportional to the energy 
supplied to it. Measurements of this were made by varying 
the current in the Wheatstone’s bridge circuit. The measure- 


ments gave the following numbers :— 


pes | 
(1) Change of resistance in 15 seconds......... — 80 | 66 103 
(Scale-divisions.) | | 


Ce Bo ce ccasesyaenc region nc rdeelnoceae [4B 99 14°7 


GB SUN teh sac aanvgall cd edogev abe ge, coe: | 3660" |.) (6:66, | ae 


during the Absorption of Electrons by Platinum. 183 


The change of resistance is proportional to the watts 
supplied within the order of accuracy of the measurements. 
The error in these measurements is considerably greater than 
that of the method adopted in the investigation, on account 
of the difficulty of determining the correction for the sensi- 
tiveness of the Wheatstone’s bridge system, which is, of 
course, different for each current used. 


§ 4, Reduction of the Galvanometer Deflevions. 


It has been pointed out that it was not possible to get the 
temperature of the grid absolutely steady. Asa rule there 
would be a regular drift corresponding to a gradual rise of 
temperature ; in other, less common, instances the drift 
would be in the opposite direction. Sometimes, of course, 
there would be an irregular variation of the zero, but unless 
this was small the readings then obtained were discarded. 

In order to eliminate the effect of the drift arising from 
changes in the steady temperature of the grid, the method 
was adopted of taking a large series of readings in succession, 
for equal periods of time, with the thermionic current alter- 
nately on and off. Thus the thermionic current would be 
allowed to flow into the grid for, say, 30 seconds. This gave 
rise to a deflexion of the galvanometer indicating a rise of 
temperature. The thermionic current would then be turned 
off for an equal period, and the galvanometer spot would be 
deflected back in the opposite direction. It would not, as a 
rule, reach the initial zero in this period of time, owing to 
the occurrence of the gradual drift. Again, on turning on 
the thermionic current for the same length of time, a de- 
flexion would be obtained in the original direction; but at 
the end of this period the reading would not be the same as 
in the first case. The effect of the drift will be eliminated 
if we subtract the mean of the first and third readings from 
the second ; and if we subtract the mean of the second and 
fourth readings from the third, we shall get independent 
values of the effect, from which the drift has been eliminated. 
Treating the successive readings in this way, it will be seen 
that we can obtain any desired number of individual deter- 
minations of the effect. Moreover, if we take a sufficiently 
large number, not only will the steady drift be eliminated, 
but the effects of any chance fluctuation in the temperature 
of the grid will also be obliterated. As a matter of fact, one 
could see at a glance whether the chance fluctuations were 
considerable or not, and only those readings in which they 
were insignificant were retained. 

It will be seen that this method of reducing the readings 


184 Profs. Richardson and Cooke on the Heat developed 


can be summarized by the following rules, according as the 
number of readings is odd or even :— 

1. Odd number of readings. To half the first and last 
add twice all the other odd readings. From this subtract 
3/2 times the second and last but one, plus twice all the other 
even readings. 

2. Even number of readings. To half the last plus 3/2 
times the second add twice all the other even readings. 
From the sum subtract half the first, plus 3/2 times the last 
but one, plus twice all the other odd terms. 

In each case the value thus obtained will be n—2 times 
the effect of putting on the current, if there are n readings. 


§ 5. Results of the Experiments. 


Experiments were made first with a grid cut out of 
platinum foil ‘0031 em. thick. The mean width of the strip 
was ‘041 cm. and its length was about 27 cm. Its resistance 
was 10°04 ohms at 14 C. and varied from about 13 to 15 
ohms during the experiments while the measurements were 
being taken. This increase in the resistance was due to the 
increase in the temperature of the strip caused by the thermal 
radiation from the hot osmium filaments. [or this reason 
the effect is measured not at the temperature of the room, 
but at some higher temperature, which may be calculated 
from the change in the resistance of the filaments. The 
resistance of the leads to the grid was *08 ohm, and this has 
to be subtracted from the above values in calculating the 
temperature. 

This grid was experimented with under two different sets 
of conditions. In the first set before each experiment it was 
placed in a beaker containing nitric acid and made the 
positive electrode while the acid was electrolysed. In this 
way nascent electrolytic oxygen was deposited on it for 
various lengths of time. In the other set of experiments 
the grid was made the negative electrode in the electrolytic 
cell, filled with the dilute sulphuric acid, so that nascent 
electrolytic hydrogen was deposited on it for various lengths 
of time. It has been shown by H. A. Wilson * that when 
platinum is made the positive electrode in an electrolytic 
cell containing nitric acid for a considerable Jength of time, 
and subsequently heated, it is found to give a relatively small 
emission of negative electrons, and this corresponds to a large 
value of w. Saturating the wire with hydrogen, either 
electrolytically or otherwise, is found to endow it with the 


* Phil. Trans. A, vol. cecil. p. 248 (1903). 


during the Absorption of Electrons by Platinum. 185 


property of relatively large negative thermionic emission, 
and corresponds to a diminution in the value of w. It 
therefore seemed desirable to examine whether this difference 
in the power of thermionic emission produced by saturating 
the metal with oxygen and hydrogen respectively had its 
counterpart in the heating effect under. investigation. 
Furthermore, in order to examine whether the effect 
measured depended very much on the geometrical configura- 
tion of the platinum forming the grid, experiments were 
also made with a grid consisting of platinum wire of 
cireular section ‘0012 cm. in diameter. The length of the 
wire in this grid was approximately 36 cm. In this case 
the filament was not subjected to the electrolytic treatment 
described above, the effect being measured only for the 
natural state of the metal. ? 


§ 6. Typical Experiment. 


The method of observing can best be made clear by 
describing a typical set of observations. In every case the 
current-E.M.F. curve was first determined, and this was 
usually repeated at the end of the set. (Generally speaking 
the thermionic current rose gradually during the course of 
the series of observations, so that at the end the saturation 
current would often be twice as great as at the beginning. 
It was also, as a rule, somewhat more difficult to saturate, as 
saturation is reached with a lower voltage the smaller the | 
current. In this experiment the values of the saturation 
currents are given in the following table. The first horizontal 
row gives the values of the negative voltave applied to the 
negative end of the filament. The second row gives the 
corresponding values of the currents at the beginning, and 
the third row those at the end. In the second row the unit 
of current is 5°66 x 10-® ampere, while in the third row it 
is 10°-4x10-® ampere. The current-H.M.F. curves always 
possess the same characteristics. 


Current. and Electromotive Force. 
Oe dst Z ey 4 5 6 7 8 9 11 15 94-9 
eos Oe eo tee 19). Ol BOTs SoD 849°) 862° S82) ST'T 


Po ae Fs (oe | 9-6 14-7, 199 24-6 287 S10. 325 349 


For very low voltages the current increases very slowly 
with the applied E.M.F. This is followed by a short range 
of voltage along which the current increases very rapidly 
and is very nearly a linear function of the voltage. This 


186 Profs. Richardson and Cooke on the Heat developed 


stage again ends rather abruptly, and after that the current 
increases very slowly with the applied voltage, of which, 
however, it is again an approximately linear function. The 
character of the curves can be best realized by reference to 
figure 3, which shows the values of the currents, obtained in 


Fig. 5. 


-5 4) 5 10 15 20 25 30 35 40 45 50 
Scale of voltage. 


another experiment plotted against the corresponding electro- 
motive forces. The values are shown thus 0. It will be 
seen that in this case approximate saturation is attained in 
the neighbourhood of 14 volts. After that the increase in 
the current is relatively small. In the experiment to which 
the numbers in the preceding table refer, the current was 
approximately saturated at 8 volts. 

The heating effect was next determined for a series of 
different voltages, and as a rule these were all chosen so as 
to exceed the voltage where approximate saturation occurred. 
In this way it was possible to avoid the difficulty which 
arises from the fact that when the current is varying rapidly 
with the voltage the mean potential difference through which 
the electrons tall does not correspond to that at the middle 
of the filament. 

In order to explain the method of taking the observations, 
a typical series for one particular voltage will now be given. 
_ In this experiment the range of voltage was from 0 to 8. 
That is to say, when the thermionic current was “‘ on” the 
potential at the negative end of the filament as measured by 
the voltmeter, V, was 8 volts. When it was “off” the 
potential of this end of the filament as measured by the volt- 
meter was 0 volt. The sliding contact at H was, therefore, 


during the Absorption of Electrons by Platinum. 187 


set so that when the rocking switch A was turned the 
reading on the voltmeter V changed from 0 to 8 volts. The 
balance was then tested by the method previously described, 
and if it was out, by more than two or three per cent. of the 
expected deflexion it was adjusted so as to be as near right 
as possible. But if the proportion was not greater than that 
indicated it was measured and subsequently allowed for. In 
the present instance the balance test eave 24 divisions “ with 
the effect,’ that is to say, it tended to produce a spurious 
effect which would increase the true effect by that aiount. 
This was recorded and no further adjustment was made. 
The thermionic current was next measured and found to be 
equal to 33°8 divisions, the micro-ammeter being shunted with 
ten ohms. The potential drop along the filament was then 
observed on the voltmeter V., and found to be equal to 
3°62 volts. The resistance of the grid was observed as 
15°28 ohms. The deflexions of the galvanometer G were 
now recorded as the thermionic current was turned on and 
off consecutively at the end of every 60 seconds. The 
following readings were taken :— 


OUP a6. 775 = 805 S089 uSa +82. Se 835 83 
Oto -2--- tha) 165, 1825. 184.185, 191 , 192-5. -1015 


The thermionic current was again read and found to be 
34°4 divisions. It had thus increased °6 of a division during 
the experiment, so that the mean value 34:1 was taken as 
the correct one. The balance was again tested and found to 


co) 
be 1°4 division “ with,” or one division less than at the 


‘beginning. The mean correction on account of the lack of 


balance is, therefore, 1°9 divisions to be subtracted. When 
the seventeen re eadings eiven in the table are treated by the 
rule described on page 184, the mean heating effect is found 
to be 105°84 divisions, con which we have to subtract 1°9 on 
account of the balance not being exact, leaving 103-94. 
This is then divided by the value of the thermionic current, 

34:1 divisions, giving 3°043 as the magnitude of the effect 
with eight volts. As a rule, and particularly if the ther- 
mionic current was increasing at all rapidly with the time, 

its value was observed after each four successive readings of 
the galvanometer, and by so doing a more exact estimate of 
the average value of the thermionic current could be obtained. 

Experiments in which there wasa big jump in the thermionic 
current or a big change in the balance during the course of 
an experiment were invariably rejected. 

A set of measurements similar to the above was made for 
each of the voltages tested. In the present experiment the 


188 Profs. Richardson and Cooke on the Heat developed 


voltages used were 15, 8, 11:1, 5, 24:2, in the order named. 
The corresponding values of the heating effect per unit 
thermionic current in scale-divisions are “respectively 4°98, 
3°04, 3°93, 2°07, and 10°04. It these are plotted against the 
voltage applied to the negative end of the filament, it will be 
seen that they are all practically on a straight line with the 
exception of the one with five volts, and in this case, since 
the effect to be measured is small on account of the smallness 
of the thermionic current and the voltage through which the 
electrons fall, the deviation from the line is probably no 
greater than the experimental error. It will be seen that 
the line cuts the voltage axis at a point on the negative side 
of the origin corresponding to —3°0 volts. If we confine 
ourselves to the points corresponding to the voltage for which 
the current was practically saturated (which in the present 
case comprises all the points which fall accurately on the line) 
it is clear that the mean potential difference through which 
the electrons fall wiil correspond to that at the middle point 
of the filament. Our origin of voltage should, therefore, be 
taken at a point to the right of that in the diagram by an 
amount equal to 4 of the drop along the filament, or 1°81 
volt. The voltage, therefore, which “is equivalent to that 
through which the electrons would have to fall in order to 
give rise to a heating effect equal to that part of the effect 
which is independent of the voltage will be 3:0+4x3°62= 
4°81 volts. 

Hach determination of the value of the constant @ involves 
the determination of a number of points on the diagram in 
the manner indicated. In most cases measurements were 
taken for a number of different voltages, and a line drawn 
through the series of points. In a few cases it was considered 
sufficient to obtain two concordant readings at each of two 
points sufficiently far apart, say 8 and 24 volts, and to draw 
a straight line through them, but generally speaking this 

was checked up by determining the point midway between 
them, and if all the points did not lie on a straight line the 
matter was investigated further. 

In figure 3 the results of another series of observations 
are shown diagrammatically. The abscissz represent voltages 
at the negative end of the filament. The thermionic current 
is shown thus O, and the heating effect per unit thermionic 
current thus ©. In this case again the points are seen to 
be on a straight line, which cuts the axis of abscisse at about 
—3 volts. 


7 


during the Absorption of Electrons by Platinum. 189 
§ 7. Discontinuity at High Potentials. 


An interesting phenomenon was observed when the investi- 
gation was pushed to higher potentials than those so far 
recorded. It was found, for example, that with the platinum 
grid which had been exposed to hydrogen, when potentials 
higher than 33 volts were used to measure the effect a smaller 
heating effect was obtained for a given thermionic current 
than at somewhat lower voltages. When the matter was 
examined in greater detail it was found that at a certain 
voltage there was a sharp drop in the value of the heating 
effect per unit current. In the experiment under discussion 
this took place at 33 volts. From 36 to 50 volts the heating 
effect per unit current appeared to be almost independent of 
the voltage. The sudden drop in the heating effect was found 
to be accompanied by a simultaneous discontinuity in the 
current-E.M.F. curve. In figure 3 the heating effect and 
the current-E.M.F. are plotted together. The points on the 
-current-E.M.F. curve are shown thus O, and the heating 
effect thus ©. We have not yet had time to examine this 
interesting phenomenon in detail, but two possible explana- 
tions suggest themselves. One is that for some reason or 
another when the voltage exceeds the critical value the dis- 
charge, or part of it, takes place to the part of the grid in 
the immediate neighbourhood of the glass supports, and part 
of the heating effect is conducted into the glass and does not 
make itself felt in the grid. The second, which is more 
interesting, is that the phenomenon is connected with the 
reflexion of the electrons or the emission of secondary elec- 
trons at the surface of the metal. In that case it seems quite 
conceivable that there may be a sudden increase in the 
amount of this effect at a certain potential: that the 
secondary electrons thus emitted escape from the grid with 
considerable velocities, manage to pass out of its sphere of 
action, as it were, and drift into the other parts of the field ; 
so that, for example, instead of entering the grid and ulti- 
mately passing through the instrument which measures the 
thermionic current they reach the positive terminal of the 
osmium filaments. Some such view would account for the 
simultaneous discontinuity in both the heating effect and 
the thermionic current. In order to account for the fact that 
at higher potentials the heating effect is independent of the 
voltage, we should have to suppose that the kinetic energy 
of the particles thus lost by the grid increased in a greater 
ratio than that of the potential driving them. When the 
investigation was pushed to still higher potentials it was 


190 Profs. Richardson and Cooke on the Heat déveloped 


found that an are or spark discharge took place which melted 
the grid. The melting took place inside the brass support 
carrying the filament. This may perhaps be regarded as an 
argument in favour of the first view. It is, however, impos- 
sible to settle the question until further experiments have 
been made; so far we have contented ourselves with being 
careful to employ only potentials below the critical point in 
order to insure the absence of complications arising from 
this cause. 
§ 8. Grid saturated with Oxygen. 
The results of all the experiments made with the grid of 


platinum strip after saturation with oxygen by electrolysis 
in nitric acid are exhibited in the following table : 


Ke @ a3 ‘ D ES Be tt (hes fag eet he ae 
Pe) | eta ae Reece ea ct eS z 2° ee |? 2 ao) 
i) gee bo ails | #8 (p28 | 2 | 
| ait aa) | w oo a ec ie. 
| Y gen SAD DH 2 | ii: 
| 2 | ghee | 60 | 02 | 982 | 174 ‘3153 | 684 | 82 | 124) 2 
(1) -O1 ae et LATE | 2. 10-04 ere 
Pe WAG LA aay: 13 hae a 
| 3. | 12hrs. | 30 ss eo: | 386 | avd ly 892) eis |2.0 143 | -24 
not touched (a) 5] 
4. | since last | 30 (1) “008 | 290 166 (5194 | 607 | ...... 13-2 | -22 
experiment. | | i 
| 5. | ZR) 30 | 306 | 164 a) 58 | 599 | 287 | 13-4 | -20 
| . 3g Ors. Us a pao’ sine | e (22°C.) 
ea a ae ae s |(1) 10-48] 5 Pei 
| 6. | 34hrs, | 30 | 006 | 3 a 165 (9) 16-85 | 7°99 | «+ 142 | °23 


In the second column are given the times during which the 
oxygen was being deposited by electrolysis. The electrolytic 
currents used were always of the order of ‘l ampere. The 
third column gives the interval of time during which the 
thermionic current was turned on or turned off. In the 
fourth and subsequent columns, where the numbers are pre- 
ceded by the figures 1 or 2 respectively in brackets, (1) 
denotes that the value in question was that at the beginning 
of the observations, whereas (2) denotes the value at the end 
of the observations. The fifth column gives the fall of poten- 
tial along the osmium filament due to the heating current. 


Final corrected 
value (volts) 


| & 
go 
a 


5°35 


5°62 


5°49 


during the Absorption of Electrons by Platinum. 191 


~The numbers in the eighth column represent the values of 


the effect in volts obtained by producing the line through the 
points back to where it cuts the voltage axis, and adding to 
the negative voltage at this point on the axis half the value 
of the potential drop in column five. Columns nine and ten 
enable an approximate estimate to be made of the temperature 
of the grid, and also the correction in column eleven to be caleu- 
lated. This correction will be considered more fully below. 


It will be seen that the numbers in the last column are in 
The agreement between the last five 
is very much better than “that of the first with any of the 


very 2£ 


others. 


vood agreement. 


We have, however, carefully compared the obser- 


vations and have not been able to detect anything in the 
numbers belonging to the first set which would warrant their 
The mean of the whole six observations is 5°65 


rejection. 
volts. 


to 5:51 volts. 


§9. Grid saturated with Hydrogen. 


Experiments were made with the grid of platinum strip 
after it had been saturated with hydrogen by electrolysis 


If we reject the first cbservation the mean is reduced 


of dilute sulphuric acid for varying periods of time. The 
results are exhibited in the following table :— 
is im « — = 4 man 6 a a 3 — 
Sees alo. | SE See eee | eet 
See Sel eel. | ex | ee |.62\421 88) 2s 
eee Sear A eas | es | eo) eS | aa 
S| ee * ¥: Pan ii ae |PF/2 | SS leF 
dy 12 ars 2 ilies SSS | 166. Ais oo 5:72 150i 26 5°19 
2. 20 hrs SON Poh Re aiclis Foe) IGS iG) S67 | ATT | 148.) 2b 4:25 
Running 
3. | since last | 30 ‘005 | 362 | 16:0 \2) 708). 4:67 | 15:2 | '25 | 4:50 
experiment. 
4. a 01) A ie Be S02 160 Vella sh) O14) 16732) 325 4°63 
&, i GO aes ‘i eroe |) he) (Ch) 2067 | oo), | 15°28) 25 4:99 
6. |Recharged.| 60 | ...... 342) 50 ((1)104 | 471 | 15°16) 08 | 436 
ci 2) 1h ae 324 | 50 ((1)264 | 423 | 15°14) ‘08 3°88 
8. |Recharged.| 30 | ...... 3°04 | 50 |(1)27°4 | 5°87 | 15°47] -07 553 
9, 30 | 012 | 302] 5:0 |(1)15-7 | 613| 139 | -07 | 5-79 
10. Recharged. : alate 3:02 | 50 (1)299 | 609] 144 | 07 | 575 


192 Profs. Richardson and Cooke on the Heat developed 


The numbers in the various columns represent the same 
quantities as the corresponding columns in the previous table. 
In the last five sets of experiments the precise treatment of 
the grid was not recorded, but before the experiments 
Nos. 6, 8, and 10 the filament was exposed for varying 
lengths of time to hydrogen by electrolysis of sulphuric acid. 
A glance at the last column will show that the agreement of 
the results among one another is not so good as in the case 
with the experiment when the strip had been exposed to 
nascent oxygen in nitric acid. We have not yet been able 
to trace the cause of this disagreement with certainty. The 
mean of all the ten values gives 4°85 volts for the effect. 
It will be noticed that the three last voltages are considerably 
higher than any of the others. If these are rejected, the 
mean is brought down to 4°49. There does not, however, 
seem to be any compelling reason for rejecting them, as the 
individual observations look satisfactory, and so far as we are 
able to judge, they were made under conditions similar to 
those which held while the other observations were being 
made. 

Taking a general view of the whole results which have 
been obtained so far, it would seem that saturation of the 
filament with hydrogen reduces the magnitude of the effect 
nearly one volt from the value obtained when it has been 
saturated with oxygen. 


§ 10. Platinum Wire Grid. 


As has been stated above, experiments were also made 
with a grid wound with platinum wire of -012 cm. diameter. 
The results of these experiments are exhibited in the follow- 
ing table :— 


= 2 2 = | sy soot gf ie 

a . = g Ss =e A a= Se 

ts na + i on* aS Ore DS ras o = 

See |) Ss | 22) 8%) ee lobe he one 

eeles| = |X| $4 1188 | ac | 2 eau 

5a | 22 = 22 | g3--|-2- | saa 

ZA | 5 Bei el tes pry |e 58.2) EF 
Sea a Wes 28 | Ber 

1.| so | 304 | 50 \ayis28) 622 | 118 | 06 | 589 

2. | 30 | 300 | 50 (ayis45| 595 | 114 | 06 | 562 
a | ee ee ee | es 

2. | 30 | 298 | 50 \1)2257; 669 | 1172 | 06 | 636 
4/120 | 298 | 50 4)2257| 569 | 1172 | 06 | 5:36 | 


during the Absorption of Electrons by Platinum. 193 


The corresponding columns give the values of the same 
quantities as in the previous table. The last two experi- 
ments were made with different intervals of time. In one 
set the thermionic current was turned on and off every thirty 
seconds, and in the other set every two minutes. A marked 
difference between the two values was obtained, but we are 
inclined to think that a great deal of this arises from some 
independent cause, as another experiment, made especially to 
test this point, and which is not recorded in the table, gave 
a small difference in the opposite direction. It is difficult to 
get accurate observations with intervals as long as two 
minutes, and the fact that the value 6°36 for thirty-second 
intervals is much greater than the number given by the two 
previous observations tends to shed doubt on this set of 
experiments.. The mean of all the experiments gives 5°81 
volts for the effect. If the third is rejected on account of its 
deviation from the mean, and the fourth on account of the 
time not being the same, the mean of the two is 5°75 volts. 
This is not very different from the values given by the 
experiments in oxygen. The fact that it is somewhat higher 
may perhaps be taken to indicate that there is an apparent 
increase in the magnitude of the effect when the thickness 
of the metal used is increased. 


§ 11. Graphical Treatment. 


In order to see at a glance the degree of consistency of 
the results a graphical method of exhibiting them has also 
been adopted. Owing to the variation of some of the 
conditions, such as, for example, the sensitiveness of the 
galvanometer from one set of experiments to another, change 
of temperature of the grid, etc., the heating effect for a 
given number of volts per unit thermionic current as 
measured in scale readings, does not mean the same thing in 
the different sets of experiments. It is, therefore, necessary 
in comparing the different experiments to reduce all the 
measurements to a uniform scale. This has been done by 
drawing the best line through each individual series and 
putting the value of the scale deflexion per unit thermionic 
current at an arbitrary voltage (as a matter of fact, 12 volts 
was taken) equal to some arbitrary quantity, say 4. The 
individual readings for different voltages were then reduced 
to the scale thus obtained and have all been plotted together in 
the accompanying diagram (PI. III. fig. 4). In this diagram 
the voitage is represented horizontally and the effect on the 
standard scale vertically. The -points for the grid saturated 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. O 


194 Profs. Richardson and Cooke on the Heat developed 


with oxygen are marked thus @, the points for the grid 
saturated with hydrogen thus ©; while the points with the 
platinum wire grid are shown thus @. As the diagram is. 
constructed, the points appear to fall into two groups, each 
lying very near a straight line. This is because the hydrogen 
group has been separated from the oxygen group by shifting 
its scale of voltage four units to the right so that for the 
points which fall in the right hand group the voltages are 
less than the reading on the scale of abscissee by 4 volts.’ 
The points for the platinum wire grid have been plotted 
twice, once on each of the two voltage scales mentioned. 
Thus they fall near to the line drawn through the oxygen 
points as well as to the line drawn through the hydrogen 
points. It will be seen that the oxygen points fall on the 
straight line with exceedingly close accuracy, and that the 
platinum wire points also fall on the same line with about 
the same accuracy. As has aiready been observed, the 
hydrogen points are not so consistent among themselves and 
the best line through them falls a little off the best line’ 
through the platinum wire points. When corrected for the’ 
joulage and for the temperature energy of the electrons, 
the best line through the oxygen points on this diagram’ 
gives for the mean value 5°45 volts, while the best line 
through the hydrogen points gives 5:05 volts. The value 
obtained in this way for oxygen is somewhat less and for 
hydrogen somewhat greater than the arithmetical mean of 
the observations previously obtained. But they agree in 
indicating that the effect when the filament has been satu- 
rated with hydrogen is lower than the value in oxygen. * 

It is remarkable that all the observations on the platinum. 
grid saturated with oxygen and the platinum wire grid, which. 
have been used in deducing the values given in the previous. 
table, should fall so nearly on the straight line as they are 
seen to do from fig. 5 (PI. III.). It will be recollected that 
the individual values for the effect recorded in those tables 
showed considerable variation. This emphasizes the im- 
portance of taking every observation with great care and. 
accuracy since it is clear that a small error in the position of 
the line makes a very large percentage error in the final 
value of the effect. 

The position of the points in figure 4 is corrected for the 
difference in the potential drop along the filament in the 
different experiments, but no correction has been made on 
account of the difference in the heating effect of the Wheat-. 
stone’s bridge current in the different cases. It is not 
believed that the omission of this correction has made any 


during the Absorption of Electrons by Platinum. 195 


appreciable difference to the accuracy of alignment of the 
points. 


§ 12. The Potential Driving the Thermionic Current. 


When the applied voltage is such that the current is 
saturated, the same number “of electrons per unit length will 
be emitted by every portion of the hot wire. In that. case 
the mean value of the potential which drives the current 
will clearly be the potential of the middle of the wire. In 
the stage where saturation has not been attained this will no 
longer be true, as a greater number of electrons will be 
emitted per unit length from those parts of the filament for 
which the negative potential is greatest. In that case, how- 
ever, the true average potential of all the emitted electrons 
will be obtained if the current-H.M.F. curves are analysed 
in the following manner :— 

The thermionic current per unit length of the wire will be 
different for different parts of the wire and will be greatest 
where the negative potential of the filament is oreatest. The 
potential will vary from point to point of “the wire on 
account of the difference of potential required to drive the 


_ heating current through it. If the potential at any point is 


V, we can denote the thermionic current per unit length at 
that point by /(V). Then the observed current, 2, will be 
the integrated effect of this over the whole length of the 
wire. Thus if / is the length of the wire, we shall have 


{ane 1l (“awyav 
—— 1S SSS ; . 
5 ( ) a svat Ja 


since a, the gradient driving the heating current, is con- 


stant. Hence 


ey : 
ov —_ LIO her IV) 1 


where V; and V, are the potentials at the two ends of the 


wire. If V is less than ior then f(Vo) is in general equal 


to 0 since it corresponds to a positively charged part of the 


wire, so that /(Vz) =o" x x &. We can thus obtain the 


value of /(V) over ie range by simply differentiating the 
current-E.M.F. curve with “respect to V, since this curve 


expresses 7 as a function of V. 
3 


196 Profs. Richardson and Cooke on the Heat developed 


In some cases it has been observed that for very small 
values of V the thermionic current had a small value 
independent of the voltage. In this case we have 


AV) = ijl or Avo)fS Bae 


where V is the potential fall along the hot wire. 


When V is greater than a é 


_ oO ae 


so that to obtain 7(V) as a function of V we proceed as 


follows :—First take V=V, between 0 and j OV Then the 


Ox 
value of /(V;) is equal, by what has gone before, too” Ss 
Next take 
Vi= V.= Vi + 
Then 


] Oe 


Since f(V;) has already been determined, differentiation of 
the current-E.M.F. curve enables us to determine /(V,). 
We then take 

OV 

Ow 9 


and thus deduce the value of /(V;). Proceeding in this 
way, we can obtain the values of /(V) corresponding to a 
series of constantly increasing values of V. We now multiply 
each value of /(V) by the corresponding value of V. V is 
the potential difference driving the current {(V)d« which 
originates from the element dw ‘of. the hot wire, so that it is 
clear that the average potential difference through which the 
electrons fall, corresponding to any observation where the 
thermionic current is 7, will be equal to 


fone = i) VAY) ae = ah VAV)AV. 


Vi= V,= V.+ /—— 


during the Absorption of Electrons by Platinum. 197 


The value of this definite integral can readily be obtained 
graphically if V/(V) is plotted as a function of V. An 
example exhibiting the method of applying this correction 
for the difference of potential at different points of the wire 
will now be given. 

The only set of observations which has been examined in 
this way are those obtained with the platinum grid saturated 
with hydrogen No. 7. In this case the observations for the 
current-H.M.F. are shown thus O in the diagram fig. 5, 
Pl. Il]. The best curve possible was drawn through the 
observational points and the values of the tangents of the 
angles of inclination, of the tangents to the various points of 
this curve, with the axis of voltage were measured. The 
direction of the tangents was determined by placing a mirror 
at various points on the curve in such a direction that the 
curve and its reflexion are continuous. The edge of the 
mirror is then perpendicular to the direction of the tangent. 
The tangents were measured at points a distance apart equal 
to the fall of potential (3°267 volts) along the strip as re- 
quired by the foregoing theory. From these the values of 


i . : : 
Vie J(V) were determined. The values of this function 
were then multiplied by V, the voltage at the negative end 
of the strip, and plotted on the diagram. The values of 


ee V7/(V) are shown thus ®. ‘They are seen to lie on 


OV fez 


a smooth curve passing through the origin. The values of 


i: ich 
sve), AMV 


were next found by counting up the number of squares in a 
portion of the diagram similar to that shaded and bounded 


on the top by the graph of svae i) on the bottom by 


the axis of voltage and on the sides by the two vertical lines 
a distance QV apart; where OV denotes the potential drop 
along the filament. The values of these integrals were then 
divided by the values of the current? from the strip at the 
corresponding voltages. The resulting quantities are denoted 
thus ®. They are seen to lie on a curve, the major part of 
which consists of a straight line pointing to a voltage equal 
to 1/2 of the drop of the potential along the filament. Since 
the diagram represents the voltage applied to the negative 
end of the filament, it follows that this line points, as it 


198 Profs. Richardson and Cooke on the Heat developed 


should, to the true zero of average voltage. Along the part 
of the curve which corresponds to approximate saturation 
the deviation from linearity cannot be detected. But where 
the current is far from saturation the points on it lie very 
considerably above the straight line, so that in this region a 
considerable error would be introduced by supposing that 
the average energy of the thermions emitted by the filaments 
corresponded to what they would gain if they all fell through 
a difference of potential equal to that at the middle of the 
filament. Practically all the observations which have been 
made fall within the part of this curve where the deviation 
frem the linear relation is less than the error of observation. 


§ 13. CORRECTIONS. 


(1) For the Temperature Energy of the Electrons. 

We have seen that the heating effect for zero applied 
voltage consists of two parts, (1) the part which we are 
measuring due to the difference of the potential energy of 
the electrons inside and outside the metal, and (2) a part 
which is equal to the difference between the kinetic energy 
of thermal agitation of the electrons as they are emitted 
from the bot metal and the value which that quantity would 
have at the temperature of the grid. Thus, if 6, is the 
temperature of the osmium filament and @, that of the grid, 
this part of the heating effect will be equal to 


: (6, — 8.) = es (0; a 0), 


where x is the number of molecules in a cu. cm. of gas under 
standard conditions of temperature and pressure. 
Then ne x 273 | 
= the translational kinetic energy in 1 cu. em. of Hy at 
0° C. and 760 mm. 
= 3p = 3x 16x 13°6 x 981 x 107 joule. 
ne = charge carried by $ cu. em. of H, in electrolysis 
= 4°327 coulombs. 
Let us assume, what is approximately correct, that the 
temperature of the osmium is 2000° C. above the temperature 
of the grid. Then substituting the above values, we find 


= (0-6) fe | volt. 


It is necessary, therefore, to subtract this amount from the 
observed value of the heating effect in order to deduce the 
part which depends on the change in the potential energy. 


during the Absorption of Electrons by Platinum. 199 


(2) Lhe Direct Heating Effect of the Thermionic Current. 

In addition to the effects which have been mentioned, the 
grid will be continuously heated by the current in the 
Wheatstone’s bridge circuit. When the thermionic current 
is turned on, the conduction current flowing along the grid 
at any point will be altered, so that there will be a change in 
the heating effect due to the conduction current. We shall 
now proceed to show how the rate of heat production due to 
this cause may be calculated. Suppose AB (fig. 6) repre- 
sents a length of heated metal emitting electrons, and the 


Fig. 6. 
A Q 


BO 
material of the grid is represented diagrammatically by 
the line OQ. Then at any instant a uniform current i 
arising from the Wheatstone’s bridge circuit will be flowing 
along OQ. In addition to this there will be a thermionic 
current flowing from AB into various points of OQ. Let 
the thermionic current into OQ at any point be 7 per unit 
length. Then the thermionic current into a length dz of OQ 
at the point 2 is equal to jdv. The total thermionic current 


into the grid will thus be J = (* jd«. Inthe arrangement 


that we have used part of the thermionic current flows out 
of one end of the grid and part out of the other (see fig. 1). 
There will, therefore, be a point M in the grid where the 
thermionic current contributes nothing to the value of the 
current along it, so that the current at this point has the same 
value zp, whether the thermionic current is on or off. Let us 
take this point as origin and let « denote distance along the 
strip, z being positive upwards. Let the co-ordinates of Q 
and O be 2; and — 2, respectively. Then the current along the 


grid at any point z=7 +). jde. Let R be the resistance per 


unit length of the grid. Then the rate of heat production in 
the length dz 


ave (io “jae) 25 4 i a zi “jde+ ( | jde) bade. 
e 0 <0 9 


200 Profs. Richardson and Cooke on the Heat developed 


When there is no thermionic current, the rate of total heat 
production 


£ (P21 
wey 1p Ak. 
—LO 


So that the increase, 5Q, due to the thermionic current 


Br "S UR 2 
= R| dx { 2i,{ jda t+ ( ( jr) } ¢ 
— 00 ae) e/ 0 


As an illustration, let us suppose that 7 has the same value 
at every point. Then 


dQ == Ra, a. a) 2) (ey = Tea) + 1? (ay” — Vo + Xo") t. 


We can determine the position of the point M from the 
conditions which have to be satisfied in order that there 
should be no current through the galvanometer G (fig. 1) 
when the thermionic current is turned on. For, we shall 
have that the drop of potential from M along S through the 
resistance R,, shunted by the battery circuit and the resis- 
tance R, to K, is equal to the drop of potential from M along 
S to L, so far as the thermionic current is concerned. The 
drop of potential from M to K due to the thermionic current 
is equal to : 

a ie a R,(B+ Ry) 
rf ‘arg jars P ("dey where P= Ry ie 
B being the resistance of the battery circuit. The drop from 


M to L 


cos | ” de \ “jae. 
0 0 


In addition, the potential drop due to the thermionic current 
From K to E 


ae!) 


} 


Sih) (jae = the drop from L to H= ni ” ide. 
val) 0 


In the particular case when j is constant, these two relations 
become 


Fei ; wf 5 
oy eL +77,P = jar’, or Ra,?+2Pa, = R2,) - ee 


and ; 
ja R, =ja,R;, or 2, Ry= aks. - re 


We see, therefore, that 2,/7,; = R,/R;. In the experiments 


during the Absorption of Electrons by Platinum. 201 


R,/R; was comparable with 200, so that z, is practically 
equal to the whole length / of the grid. Let us, therefore, 
put as an approximation #=/ and #,/%)=0. So that 
6Q = RI(477/?—wWl). In this case we also have Ri = 8, the 
resistance of the grid, and j] = J. So that 


Sey etd i). 


In the experiments the maximum values of these quantities 
were respectively S=15 ohms, J =38X10-* amp., and 
7=17x10-* amp. So that the maximum value of this 
correction §Q/J = °25 volt. 

Although the above calculation of the correction has only 
been carried out for the case where j is constant, it can 
readily be seen that a similar result will foliow for any 
distribution of 7 which is symmetrical about the central point 
of the grid. As this condition was very nearly fulfilled in 
practice, the above method of calculation has been used in 
estimating the magnitude of the correction for the direct 
heating effect of the thermionic current. The relation 
between the direction of the thermionic current and the 
direction of the bridge current was such that this correction 
involved a deduction from the observed value in all cases. 
The actual values of the amounts which have to be sub- 
tracted on this account are given in the last column but one 
of the various tables above. 


§ 14. Possible Sources of Lrror. 

It will readily be conceded that the foregoing results leave 
no doubt as to the existence of the effect under investigation 
or of its order of magnitude ; but, at the same time, it is 
very difficult to obtain results of a high order of accuracy. 
We are unable at present to locate the source of such 
inconsistencies as have been experienced. A glance at the 
tables would seem to indicate that there is a correlation 
between the magnitude of the observed effect and the 
pressure of the gas in the apparatus. If it is desired to 
carry out experiments of this nature with reasonable rapidity 
it is difficult to reduce the pressure of a gas below ‘005 mm. 
owing to the fact that the development of heat in the osmium 
filament raises the temperature of the whole apparatus very 
considerably and causes an inconvenient amount of gas to 
be evolved. We have made direct experiments in order to 
see if an increase of gas pressure gave rise to an increase in 
the apparent value of the effect. These experiments show 
that if there is any effect due to pressure it is probably too 
small to account for the observed differences. 


202 Profs. Richardson and Cooke on the Heat developed 


Another difficulty that we have to contend with arises 
from the fact that the hot filaments continually sputter 
particles on to surrounding objects, so that the grid very 
rapidly becomes covered with a laver of material “deposited 
on it from the osmium filaments. Weare not sure, however, 
that this alters the magnitude of the observed effeah as in 
some cases, for example oxygen No. 4, the value 5°58 was 
obtained after a continuous heating of some twenty hours ; 
whereas the preceding experiment gave a value of 5°62. In 
other experiments a change seemed to be observed after 
continued heating, but there is no conclusive evidence that 
the change was ‘due to the sputtered material. The possi- 
bility of an alteration both from the pressure of the gas and 
from sputtering lands us in a dilemma, because to get the 
apparatus down to a really low pressure it 1s necessary to 
heat the filaments continually for a long time, whereas to 
avoid the accumulation of the sputtered material it would be 
advisable to take the observations as quickly as possible after 
the apparatus had been set up. 

We have already pointed out that the rate at which the 
grid heats up depends not only on the total rate of heat 
production within it, but also on its mode of distribution. 
For instance, for a given rate of heat production the increase 
of temperature in a given interval is not the same when the 
heat is produced throughout the volume of the grid by 
increasing the current in the Wheatstone’s bridge cireuit 
as it is when it is produced by the impact of the electrons. 
A source of error of somewhat similar character may 
possibly arise when the heating effects at different voltages 
are compared with one another by comparing the rise in 
temperature in equal times; since it is probable that the 
distribution of the current into the grid will be different at 
different voltages. We should expect that’ the electrons 
would be more likely to be collected into a small region in 
the centre of the grid when the difference of potential 
between the filaments and the grid is relatively large. This 
might introduce a difference in the rate of rise of tem- 
peratare of the grid as between high and low voltages, even 
if the actual total rate of heat production were the same. It 
is difficult either to test for this effect or to eliminate it if it 
occurs, but it seems fairly certain that it cannot lead to very 
big errors. It is to be borne in mind that since the resistance 
of the grid is proportional to the absolute temperature at 
every point, the increase in the total resistance will be inde- 
pendent of the distribution of the heat communicated to 
it, provided that the total amount of heat communicated is 


during the Absorption of Electrons by Platinum. 203 


identical in the cases compared. It is also necessary that the 
heat communicated should be similarly distributed about 
the cross section of the grid and that the latter should be 
uniform. 

One possible source of error that we considered arises from 
the cooling effect produced by the escape of the thermions 
from the osmium filament. This will make the temperature 
ot the filament lower when the thermionic current is on 
than when it is off, and hence the thermal energy radiated 
to the grid will be less when the thermionic current is on. 
We should therefore expect a change of temperature of the 
grid independently of any of the causes discussed hitherto. 
This effect was tested for in the following manner. 

The grid and its connexions, which were otherwise insu- 
lated, were connected by a wire with the negative end of the 
filament. A suitably high potential difference, which could 
be reversed, was applied between the filament and the sur- 
rounding brass box, so that the thermionic saturation current 
could be made to flow at will from the filament to the box 
when desired. With this arrangement there is no thermionic 
current from the filament to the grid, so that any change 
produced in the temperature of the latter will arise from 
changes in the thermal radiation it received from the 
filaments. Under these circumstances it was found that no 
change was produced in the resistance of the grid when the 
thermionic current from the filaments to the case was turned 
on or off. This shows that under the conditions of the 
experiments the cooling effect arising from the emission of 
the electrons by the filaments is too small to exert any 
appreciable influence on the temperature of the grid through 
the change in thermal radiation which it causes. This 
possibility can therefore be entirely left out of consideration 


as a disturbing factor. 


§ 15. Comparison with the Work done during the emission of 
| Electrons from [ot Metals. 

Without being able to assign any very satisfactory reason, 
we are inclined to think that the most probable values of the 
effect are those which are obtained after the high values in 
the tables are omitted. Thus for platinum saturated with 
oxygen by electrolysis in nitric acid the value of ¢@ probably 
corresponds very closely with the work done in falling 
through a potential difference of 5°5 volts, whereas for 
platinum saturated with hydrogen by electrolysis of dilute 
sulphuric acid the mean value obtained, when the last three 
series of observations are neglected, is 45 volts. 


204 Profs. Richardson and Cooke on the Heat developed 


On account of the greater consistency of the measurements 
the value for oxygen is probably considerably more reliable 
than that for hydrogen. With the exception of the single 
high value, which has been omitted, the five remaining 
determinations for oxygen agree w ith one another within 
the limits of observational error. 

It is interesting to compare these values of @ with the 
values of the corresponding quantity deduced from experi- 
ments on the variation with temperature of the negative 
thermionic emission from hot platinum. This method of 
deducing the value of ¢@ has already been explained by 
O. W. Richardson. A _ little further explanation will 
perhaps not be superfluous. The coefficient 4 in the formula 
A@"?e—/8, which represents the variation of the thermi- 
onic emission from a hot metal with the temperature, 
is ad where ¢ is the charge on an electron and Ris the 
gas constant reckoned for a single molecule. Hence if n is 
the number of molecules in 1: c.c. of a gas under standard 
conditions of temperature and pressure, 


where R, is the constant in the equation pu=R,@ reckoned 
for 1 c.c. of gas at 0° C. and 760 mm. pressure. ne is clearly 
the quantity of electricity required to liberate half a cubic 
centimetre of H, at 0° C. and 760 mm. pressure in a water 
voltameter, and is =*4327 electromagnetic units. The value 
of R, is 3:72 x 10°erg/°C. Substituting these values, we find 


— @ = 8°59 x 10-°x b volts. 


The value of @ given by Richardson®* in his first series 
of measurements is 4°] yolts. This is a little lower than 
what the value of 6 from which it is calculated requires, on 
account of approximate values of R and e having been used. 
Recalculating from the experimental value of }, using the 
relation given in the last paragraph, we find = 4-26 volts. 

More recent work has shown that this value is smaller 
than that from pure platinum, as the results there given 
indicate that the metal used in these experiments was probably 
not free from traces of hydrogen. ‘The best value hitherto 
obtained is probably one given by H. A. Wilson f for a wire 
carefully freed from hy drogen by treatment with nitric acid. 


* Phil. Trans. A, vol cci. p. 497 (1903). 
+ Phil. Trans, A, vol. ecii. p. 248 (1903). 


during the Absorption of Electrons by Platinum. 205 


His value of 6, when reduced in the manner described above, 
gives 6=5'63 volts. Another satisfactory set of measure- 
ments is given by Deininger*, who finds ¢$=5:1 volts. 
Deininger, however, seems to have used the same approximate 
values of R and e as were used by Richardson in his first 
paper. Using the better values of the constant e/R given 
above, we find from Deininger’s value of b that 6 ==5°26 volts. 
Another determination to which some weight should be 
attached is a more recent one by Richardson +. The indivi- 
dual observations in this set are not so good as in the two 
others, but they have the merit of comprising concordant 
values for two different specimens of platinum. 

If we take the mean of the values given in the last 
paragraph, assigning different weights to the different deter- 
minations, namely 3 to Wilson’s, 2 to Deininger’s, and 1 to 
Richardson’s, we find ¢=5°54 volts. This may be taken to 
be coincident with the value 5°5 volts found’ for the corre- 
sponding quantity in the present investigation, within the 
limits of the errors of observation. 

Values of } for the electronic emission from hot platinum 
in an atmosphere of hydrogen have been given by Wilson ¢ 
and Richardson §. These depend not only on the pressure 
of the hydrogen, but on the previous history of the platinum 
as well. The values given by Wilson are for a new wire 
and are as follows :— 


Pressure of H... ‘0013 mm. “bi2 mm... 1£33.mm. 
Walne of 6.2.2. 6°0 x 104 4°30 x 104 1°80 x 104 
Malue ob @ 2.0.2. 5°16 volts 3°70 volts 1-546 volts. 


Richardson’s values, which are for an old wire in hydrogen, 
are as follows :— 


Pressure of H... 1°9 mm. 226 mm. 
Valne ot 6 ...... 6°0 x 10° 21336 10" 
Walne: afd... . 5°16 volts 2°39 volts. 


The platinum in our experiments was saturated with 
hydrogen by electrolysis of dilute sulphuric acid and it is 
impossible to say which of the above values our results should 
be compared with. All the results in hydrogen have one 
feature in common, namely that they give a smaller value of 
@ than that given by a platinum wire which is free from 
hydrogen. 

* Ann. der Phys. iv. vol. xxv. p. 304 (1908). 
f+ Phil. Trans. A, vol. cevii. p. 1 (1906). 
t Loe. cit. § Loe, cit. 


206 Dr. S. W. J. Smith on the Weston Cell 


§ 16. The Concentration of the Free Electrons in Platinum. 


The fact that the values of ¢, given by measurements of the 
thermionic emission and by the heating effect, are so nearly: 
identical would seem to indicate that it does net depend very 
much on the temperature of the metal. It has been pointed 
out by Richardson* that the temperature variation of @ is of 
importance in connexion with the determination of the con- 
centration of the free electrons in a metal. ‘The coincidence 
of the two values of @ would indicate that the value of the 
concentration deduced from experiments on the thermionic 


emission is not far from the truth. 


17, Conclusion. 


Further measurements on a series of metals are in progress 
and the results will be published shortly. It seems desirable 
to postpone further discussion of the results of the present 
paper until the whole can be considered together. 

In concluding we wish to express our thanks to Messrs. 
Baldwin, Carter, Critchlow, Ferger, Frederick, and Gibbs, 
Honours Students in Physics, who have assisted us in taking 
a number of the observations. 


Palmer Laboratory, 
Princeton University. 


XV. The Limitations of the Weston Cell as a Standard of 
Electromotive Force. By 8S. W. J. Smuiru, J.A., D.Sc., 
Lecturer on Physics, Imperial College of Science and 
Technology F. 

§1. HIS paper contains an attempt to explain Mr. F. 

E. Smith’s recent experiments on the cadmium 
amalgams of the Weston cells {in terms of the theory of 
solutions. 

The manner in which, according to this theory, the amal- 
gams crystallize is indicated, and it is shown why, with this 
mode of crystallization, the effect of ihe slowness of diffusion 
is so pronounced, and also why sudden cooling to a tempe- 
rature below the freezing point of mercury must produce a 
comparatively uniform alloy. 

The cause of the most obvious differences between the 
behaviour of the “chilled” and of the “slowly cooled” 
amalgams is then at once apparent. 

* Phil. Trans. A, vol. ccii. p. 543 (1903) ; Phys. Rev. vol. xxvii, p. 528 


1908). 
+ Communicated by the Physical Society : read May 27, 1910. 


{ Phil. Mag. Feb. 1910, pp. 250-276. 


Po - le ot ee a re 
oe? ihe 
ee ~ : 

aunt 


as a Standard of Flectromotive Force. 207 


But there are certain much less obvious differences of 
which, in view of the importance of the Weston cell as a 
standard (if for no other reason), it is desirable to know the 
cause. 

An all-fluid amalgam, of course, yields the same E.M.F. 
whether previously chilled or cooled slowly. An all-solid 
alloy, on the other hand, always gives a higher E.M.F. when 
solidified by chilling. 

The E.M.}’. of the chilled amalgam thus in general equals 
or exceeds that of the slowly cooled amalgam. 

But in the range of the two-phase alloys, over which the 
E.M.F. varies comparatively little with the total percentage 
of Cd, the opposite is true. The E.M.F. of the slowly-cooled 
amaloam now either equals or exceeds that of the chilled 
amalgam. 

The excess never amounts to more than a few hundred- 
thousandths of a volt ; but is important in measurements of 
the highest precision and requires explanation. 

It is shown that this phenomenon may be due to electro- 
lytic surface effects arising from the lightness and want of 
uniformity of composition of the solid grains in the slowly 
cooled amalgams. 

Instances of similar effects, liable to escape notice, are to 
be found amongst the data for the all-solid alloys. 

Finally, the question as to whether there is any range 
over which the E.M.F. is absolutely independent of the 
percentage of cadmium is discussed. 

Theory and experiment alike suggest that the E.M.F. 
must rise as the percentage increases ; but the variation 
frequently does not amount to more than a few millionths 
of a volt for one per cent. variation of the cadmium content. 


§2. A mode of crystallization of binary alloys.—The 
various ways in which fluid mixtures of two metals can 
freeze have been carefully studied within recent years. One 
of these is shown in fig. 1. The abscissee represent per- 
centages of one metal (B) in the mixture, reckoned from a 
zero at which the other metal (A) alone is present. The 
ordinates represent temperatures. The melting point of A 
is O. The “ freezing point curve” 6,va gives the tempera- 
tures at which different alloys begin to solidify, and shows 
that the freezing point rises continuously as the percentage 
of B in the mixture increases. The “melting point curve” 
Ayb gives the composition of the solid which deposits from 
any particular liquid when it begins to freeze. Thus acooling 
fluid containing w per cent. of B begins to freeze at @ and » 


208 Dr. S. W. J. Smith on the Weston Cell 


is the percentage of B which the solid first deposited con- 
tains. At any given temperature (@), liquid and solid alloys 


can exist in equilibrium only when their percentage composi- 
tions have definite values (represented by w and y respectively). 
The thermodynamical method of accounting for this condition 
of equilibrium is referred to later (§§ 12 and 14). 
Roozeboom was the first to suggest that the thermal 
variation of the constitution of cadmium amalgams may be 
determined by curves like those of fig. 1, so long as the 
percentage of cadmium does not pass a certain limit (not 
exceeded in the experiments discussed below), and this 
suggestion was found to accord with various experimental 


data obtained by Bil. 


§ 3. Recent experiments on cadmium amalgams.—Mr. F. E. 
Smith has thrown fresh light upon the problem and provided 
new material for investigation by examining the effect of 
“chilling,” ¢. e. of cooling the amalgams suddenly from 
temperatures at which they are wholly fluid to a temperature 
below the freezing point of mercury. 

The most striking result of his experiments is shown in 


as a Standard of Hlectromotive Force. 209 


fig. 2, which exhibits (diagrammatically) how, at a constant 
temperature, the E.M.F. of a Weston cell alters with the 


Fig. 2. 


FLECTROMOTIVE FORCE. 


FERCENTAGE OF CADMIUM. 


percentage of Cd in the amalgam. In one set of experi- 
ments, represented by the curve ABCD, the amalgam was 
cooled suddenly (as above) to —50° C. and then allowed to 
rise in temperature to 0° C. before the cell of which it formed 
part was constructed. In the other set, represented by ABEF, 
the amalgam was cooled “slowly ”’ (2. e. from the liquid state 
to 0° C. in several hours) before being used. 

On the scale of representation, the curves are identical 
from A to C, and BCE is a horizontal straight line. The 
observed time effects and variations in the horizontal parts 
of the curves are discussed later. 

It is easy to anticipate from fig. 1 that the curve for 
measurements like those of fig. 2 will consist of two branches 
Joined by an intermediate horizontal portion. For ata given 
temperature 0, any amalgam containing less than « per cent. 
Cd would be all-liquid,-and any amalgam containing more 
than y per cent. Cd might be all-solid ; while intermediate 
amalgams could consist of mixtures in different proportions 
of w per cent. liquid and y per cent. solid respectively. 

In all-liquid and all-solid amalgams the electromotive force 
would, it is natural to suppose, vary continuously with the 
percentage of Cd. In mixtures consisting of the same two 
constituents (in different proportions) it might similarly be 
expected that the electromotive effect would remain constant. 

Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. P 


210 Dr. S. W. J. Smith’on the Weston Cell 


In this way the general form, either of ABCD or of 
ACEF, could be Mecano’ for ; ae the cause of the dif- 
ference between the two curves is not immediately obvious. 


$4. The freezina of slowly-cooled amalgams.—To under- 
stand the exact significance of fig. 2,1t 1s necessary to consider 
how a mixture, to which fig. 1 applies, freezes. 

According to this figure, an alloy containing a per cent. 
Cd should begin to freeze at 6; and should apparently become 
solid at 6... But, because of the extreme slowness of diffusion 
in solids, this will happen only when the rate of cooling is so 
slow that it cannot be dealt with in practice. 

At the temperature 6, a fluid amalgam containing a per 
cent. Cd and a solid amalgam containing 0 per cent. Cd are 
in equilibrium. When the temperature _ is lowered slightly 
a fluid amalgam containing slightly less than a per cent. Cd 
will be in “equilibrium with a solid amaleam containing 
slightly less than 6 per cent. A small quantity of the 
a per cent. amalgam may therefore solidify. 

As the temperature falls the percentages of Cd contained 
by fluid and solid amalgams in equilibrium become continu- 
ously lower. ‘The amount of solid material will therefore 
increase ; but the percentage of Cd in the newest crystals 
will alwe ays be less than in those previously formed. 

The fluid existing at any given stage of the cooling will 
be in equilibrinm with the solid with which it is in direct 
contact, but, since most of the successive growths will take 
place around earlier erystals, this solid will ‘in general enclose 
older solid, richer in Cd. 

There must thus be a continuous diffusion of Cd in the 
erystallized part of the material from within towards the 
surface. 

In any practical case, where the rate of cooling is not 
infinitely slow, the diffusion outwards will be very gradual 
and will not keep pace with the lowering of temperature. 

Thus, although (for true equilibrium) an amalgam con- 
taining a per cent. Cd should be all-solid just below @, a 
considerable quantity of liquid, containing ¢ per cent. Cd, 
will remain. The solid in contact with this liquid will contain 
a per cent. Cd, but will envelop a considerable quantity of 
solid richer in Cd than itself, and there will be, in conse- 
quence, neither so little ¢ per cent. liquid nor so much a per 
cent. solid as true equilibrium would imply. 

A superior estimate of the amount of liquid remaining at 0, 
can, however, be found. Thus we may imagine that the 
cooling from 0; to @,, of an amalgam containing a per cent. 


bo 


as a Standard of Electromotive Force. 2il 


Cd, takes place by steps of d@, and that no diffusion takes 
place from the solid formed in one step to that formed in the 
next. Also that the solid formed in any step is of uniform 
composition and in complete equilibrium with the liquid 
which remains. Under such conditions, it can be estimated 
that (in some of the amalgams) about one third of the material 
might still be fluid at @, although, according to fig. 1, the 
whole should be solid *. 


§ 5. The effect of sudden cooling.—F rom the above sketch 
of the process of crystallization we see that the surface of 
a “slowly ” cooled amalgam will generally contain a lower 
percentage of cadmium than the material as a whole, and 
may even be fluid, although the temperature and percentage 
composition of the material are such that it should (in true 
equilibrium) be a uniform solid. 

We see also that the relation between the curves ABCD 
and ABEF of fig. 2 is immediately explicable if, for any 
reason, the amalgams of the branch CD are of more uniform 
composition than those of HF. 

The former amalgams were cooled suddenly from the fluid 
state to a temperature much below that at which they would 
have become completely solid if the rate of cooling had been 
infinitely slow. Hach alloy would therefore pass rapidly 
through the range of temperature in which equilibrium 
between two phases is possible, and although, in each element 
of the material, there might be incipient crystallization with 
accompanying redistribution of the Cd, as the temperature 
fell, this process being slow could not proceed very far. 

The greater part of the solidification would thus take 


* Thus at the end of the first step the temperature is 0, —66, the 
liquid phase contains (a—da) per cent. Cd and the solid phase (b—4d) 
per cent. And, of m grams of a per cent. alloy, the quantity 

om, = (17 Reo lyme m 

+ (6—a)—8(b—a) 


will have frozen. It happens that for a considerable range of tempera- 
tures and concentrations in the present case, the liquidus and solidus 
curves of fig. 1 are sufficiently nearly parallel straight lines to permit 
the assumption, (6—a)=0, between 6; and 6,. From this also, if we 
assume that there are n equal steps of 60 in the cooling process, we get 
ndu=b—a, and hence 6m=m/n. The quantity of liquid remaining at 


the end of the first step is thus m(1—-). Continuing the process it 
will be found that the quantity of liquid remaining at the end of the nth 
step is m(1 ss which, if we assume » to be very large, has the value 


m/2°72 very nearly. : 
2 


he Dr. S. W.-J. Smith on the Weston Cell 


place ata temperature at which Cd and Hg ean exist to- 
gether in equilibrium only as a homogeneous” mixture ™, Ut 
would therefore occur without redistribution of the Cd with 
respect tothe Hg. What “ differential” crystallization there 
was would be on a scale relatively so minute that the process 
of equalization by diffusion and the approach to true equili- 
brium would take place comparatively rapidly as the tempera- 
ture rose. 

There is, therefore, no an in finding a satisfactory 
general interpretation of fig. 2, as a consequence of fig. 1. 

We may now proceed to examine the data more minutely. 


§ 6. Quantitative comparison of the “ chilled” and “ slowly 
eooled”? amalgams.—It is reasonable to suppose that two 
amalgams have the same surface composition when they give 
the same steady HE.M.F’. at the same temperature, even 
although their average compositions may be different. 

Hence, if we assume as a first approximation that the 
values of « and y at any temperature @ (when fig. 1 is taken 
to represent cadmium amalgams) can be deduced from the 
thermo-electromotive properties of the series of chilled 
amalgams of Table VIII. (/. ¢. p. 268), we can determine 
the state of the surface of any slowly cooled amalgam when 
the E.M.F. which it gives is known. 

For this purpose, some of the data for the slowly-cooled 
amalgams (Tables I. and IL., 1. e. pp. 256, 257) were plotted, 
as in fig. 3, along with the ‘data for the chilled amalgams. 
The dotted curves refer to the former. 

Table A, below (p. 214), summarizes the inferences which 
can be drawn from the figure when the data are interpreted 
in the way above described. 

Hach row of numbers gives, for a particular temperature, 
the surface constituent or constituents of the slowly cooled 
amalgams (11 to 20 per cent. Cd) deduced by interpolation 
from the curve, at that temperature, for the chilled amal- 
gams. The table begins with the results at 15° C. after the 
amalgams had stood for three months at this temperature. 

(i.) At 15° C. (according to Table VIII.) any amalgam 
containing less than about 4°3 per cent. Cd should be all 
liquid, and any containing more than about 12°85 per cent. 
Cd should be all solid (if in equilibrium). According to 
Table A, the surface of the 11 per cent. slowly cooled 
amalgam i is a mixture of these alloys. The surfaces of the 
12 per cent. and 13 per cent. amalgams appear to be solid 
alloys containing slightly above 12°85 per cent. Cd. Hach of 


* See § 14 below. 


ce 


Fig. 3. 


20 --_—-— 


Dr. 8. W. J. Smith on the Weston Cell 


214 


ao oe | | | ae ae (‘xouddu) 


quege ae) 89-8 | 10-80 | 8-91 | GL | GEL éSP-EL | 6G-IT | B0-1T | 10-11 O-IT 


‘TA WOay 


e 
SS —= | 
‘OQ oGP wos ((xoidde) (xorddv) | 2 | Peeupep 
Suljooe 19} LS Les 
skup gynoqe} of CO-S1.| Sell Zeb! er-eb PSS tl | Geet | G0-tl eda t ) 
3 | | 0-11 | OTL 
= (‘xoudde) (-xouddu)|(‘xoxdde) | ass 
ob | $9-8T | S827 829.97 8-8 8-8 8:8 8:8 8-8 8-8 8:8 “A 
Gy eel 
pace | éG-91 | 6G-9T | 6-91 | éS OT | 29-91 | 2¢-9T | 6¢-9T "LITA pur 
| ———|}---—}- ae SS a | ee TE “IL “LT S°14&, 
(xeiddr) (xoudde)) (*xo1ddv) WO) paonpep 
OF | 89-8L | S641 | 8.91 6-L 6-2 6-1 6-1 6-2 6-2 OL | ear 
oak GCL | 6SL| 6ST] 6SE| GST | 6c | 6ST 
Oe ee eee -—_——_ | sanzeaedinay, quatayip 
(xoidde) (‘xoidde)|(‘xoudde) (‘xoadde)| (‘xoidde) <« fs 
ro) . op 6L, L, | PT : : : ‘ 7-9 HY 
OG | SHST | ELL | GOL | gh OT | SAF | SLOT | PLOT | OLPT | CLT) | agremy us 
ee ee a | ee ee oS a eee POJ° 
(‘xoadde) (‘xoadde)} (-xordde) sosujusd1ag 
006 | 89-81 | 8621 | 6 9L | GL-SL | 98-81 | 24-81 | 9-8L | 6S-SI | 6S-EL =e Th ees 
G.e 
Se 2 ee (-xoadda) (-xoudde) (‘xoaddr) = 


‘O oGL 32 


S{{}UOUL 


goge ine | o8T | S9-8T | G6-LT | 88-YT | S4S1 | 26ST | SET | TEL] 66T | 88ET | oon, | t 


4 | <4 


"aed VUy Ul 


a "HAIN YR - : 
L 0d 6T ‘ST “LT "OT ‘GT ‘TI or “OL “Et PO Je 


-adwiay, Q 
eSvqUcddeg Uo] 


ee a ee ee eee 
| ‘VY F1dV], 


as a Standard of Electromotive Force. 215 


the succeeding amalgams (with the exception of the 16 per 
cent. amalgam which for some accidental * reason behaves 
irregularly) is superficially weaker in Cd, in nearly the same 
proportion (0°92 to 0°94) in each case, than the amalgam as 
a whole. 

(ii.) At 20° C. (to which the cells were next heated) any 
amalgam containing less than about 4:9 per cent. Cd should 
be all-liquid, and any containing more than about 13:5 per 
cent. Cd should be all-solid. The surface of the 11 per cent. 
amalgam is a mixture of these alloys, and as before the 
surfaces of the 12 per cent. and 13 per cent. amalgams 
appear to be solid and to contain slightly more than the 
percentage of Od (13°5 per cent.) in the richer constituent 
of the two-phase amalgam. The surface of the 14 per cent. 
amaleam contains about 13°6 per cent. Cd. 

It will be noticed that the surface concentrations of the 
richer alloys (15 to 20 per cent ) as deduced by this method 
are, as nearly as can be measured, the same at 20° as at 15°. 

Giii.) At 30° C., the equilibrium amalgams should (accord- 
ing to Table VIII.) contain approximately 6°4 per cent. and 
14-7 per cent. Cd. With certain limitations, discussed later 
in § 9, (which apply equally to similar cases at the other 
temperatures), the surface of the 11 per cent. amalgam is 
now a mixture of these alloys. And we might expect that 
the surfaces of the 12 to 16 per cent. amalgams would be 
similarly constituted. But, as in (i.) and (ii.), the surface 
film in these amalgams seems to be solid, since it contains a 
slightly higher percentage of Cd than the solid component 
of the two-phase system. 

From the results in column (ji.) we might expect the 
surfaces of the 17 to 20 per cent. alloys to remain unchanged. 
It will be seen that this is true of the 19 and 20 per cent. 
amalgams ; but the percentages of Cd in the surtaces of 
the 17 and 18 per cent. amalgams are beginning to diminish 
slightly and are exhibiting a time effect. 

The significance of these unlooked-for differences is 
discussed in § 10 below. 

Gv.) At 40° C. the limiting alloys contain about 7°9 per 
cent. and 15°9 per cent. Cd. As we might expect, the 
surfaces of the 19 and 20 per cent. amalgams remain un- 
changed. On the other hand, the percentage of Cd in the 

* It is obvious that irregularities of cooling and distribution must 
sometimes occur. That sudden changes of the slopes of the curves, 
between i5 and 20 per cent. Cd, are due to accidental irregularities is 
shown by comparison of the present data with those for another series 
(Table VI. 7. c.), in which similar changes of slope occur at other per- 
centages, ais a 


216 Dr. S. W. J. Smith on the Weston Cell 


surface of the 18 per cent. amalgam is continuing to fall. 
The surface of the 17 per cent. amalgam has become 
practically a two-phase system. 

(v.) At 45° C., the limiting percentages are about 8°8 and 
165. The surface of the 20 per cent. amalgam remains 
practically unchanged; but, unexpectedly as before, the 
surface percentage in the 19 per cent. amalgam is now 
slowly falling. The decrease at the surface of the 18 per 
cent. amalgam continues. 

(vi.) The amalgams were now cooled to 0° C. After six 
weeks at this temperature (last row of Table A) the surfaces 
of all the amalgams containing more than 14 per cent. Cd 
(excluding the irregular 16 per cent. ama!gam) had returned 
practically to the state in which they were when the measure- 
ments at 15°, as in (i.), were made. 

Table IX. (1. c. p. 269), described as typical, can be 
analysed (by comparison with Table VIII.) in exactly the 
same way as Tables I. and Il. In connexion with this 
Table the author writes :—‘“ At temperatures near to but 
below the first transition temperature the diffusive processes 
in an unstable amalgam are no doubt accelerated, and the 
outer shell becomes richer in cadmium with a corresponding 
increase in the E.M.F. of the cell..... ” No doubt rise of 
temperature will accelerate the diffusion ; but it is to be 
remarked that, as shownin Table A, an “ unstable” amalgam 
which has stood for some time at the ordinary temperature 
can be raised through 20° or 30° C. without measurable 
increase in the surface percentage of cadmium, and that 
the first noticeable effect of temperature rise is a surface 
decrease of an unexpected kind. The same effect is shown 
in Table LX. 

Sometimes (e. g., J. ¢. pp. 260, 261, and Conelusion 1, 
p- 274) Mr. F. E. Smith writes as if an unstable amalgam 
may consist of one central solid mass surrounded by a shell 
of lower concentration. It seems to me, however, that this 
state of affairs must be very exceptional. 


§7. Lhe probable structure of frozen amalgams.—lIt is a 
well-established characteristic of erystallization in general, 
that it proceeds around nuclei distributed more or less 
uniformly throughout the cooling material. The uniformity 
of distribution of the nuclear growths will be affected in the 
present case by the fact that the solid grains are of less 
density than the fluid out of which they separate. The first 
grains to form must tend to rise. On account of their 
smallness they will not rise rapidly. As crystallization 


{ea 


as a Standard of Electromotive Force. 217 


proceeds a solid net-work will develop within the material. 
The later crystallization will take place from fluid entangled 
within the meshes of this net-work. Consequently the 
material need not in general separate into upper, all-solid, 
and lower, all-fluid, parts. . 

In an amalgam which is still partially fluid, but would be 
all-solid in true equilibrium, at the temperature of obser- 
vation, the mean concentration of a layer near the surface 
will no doubt be greater than that of a Jayer near the bottom. 
But the difference need not be very marked, and there may 
be an appreciable quantity of fluid in the spaces between 
individual grains near the surface, which will take a long 
time to disappear. If the density effects are appreciable, a 
greater proportion of the earliest formed crystals will be 
present in the upper layers. The centres of the grains 
nearest the surface may then be richer in Cd than the centres 
of those lewer down ; but even when the amalgam has stood 
long enough for the upper portion to become solid through- 
out, there will be graduations in the percentage of (d, from 
point to point in that portion, of which the existence will 
become obvious (as described below) when the temperature 
is raised. 


§ 8. An effect of the presence of the electrolyte at the surface 
of the amalgam.—tThe surface layer of a partially crystallized 
amalgam will consist of a number of grains between which 
are spaces filled with liquid amalgam. Some of the solid 
grains will in general project slightly above the mean surface 
level. The layer of fluid amalgam covering these will be, at 
most, very thin. 

Under the ordinary process of diffusion from the grains 
the liquid surrounding them will gradually diminish in 
amount, more or less uniformly in all directions. But it is 
important to notice that the process of equalization of distri- 
bution of Cd will be accelerated, in the surface, when (as in 
the case of the Weston cell) the amalgam is covered by a 
solution of cadmium sulphate. 

The very thin layers of x per cent. fluid in immediate 


_ contact with the surface grains * will soon receive enough Cd 


by diffusion to convert them into solid containing more than 
y per cent. Cd (fig. 1). In consequence they will no longer 
be in electromotive equilibrium with the neighbouring, 
relatively thick, layers of surface fluid which still contain 


* When a saturated solution of CdSO, is poured over the fresh 
surface of a partially fluid amalgam, the positions of these surface grains 
can easily be seen. 


218 Dr. S. W. J. Smith on the Weston Cell 


only # per cent. Cd. Electrolytic action will ensue. Cad- 
mium will enter solution round the surface grains and will 
be deposited upon the adjacent fluid. 

The electrolyte will thus act as a distributor, over the 
whole surface, of the cadmium diffusing from the surface 
grains. The surface will thus tend to acquire a thin and 
probably solid skin which, on account cf its lesser density, 
will have no tendency to sink below the underlying flnid. 

In the presence ‘of CdSO, solution, the ‘whole of the 
surface film may therefore become solid and of uniform com- 
position, although, on account of the extreme slowness of 
ordinary intermolecular diffusion, there may still be an 


appreciable quantity of fluid alloy underneath. 


§ 9. “ Minor irregularities” of Weston cells —Regarding 
the surface behaviour of partially fluid amalgams — in the 
above way, it is possible to account for peculi arities, at first 
sight perplexing, ot the horizontal branches of the curves for 
the chilled and the slow ly cooled amalgams. The vertical 
scale of fig. 3 is not sufficiently open to show these peculiar- 
ities clearly. They are exhibited in fig. 4 which represents 
the behaviour of the amalgams, at various temperatures, over 
the range represented by the branch BC of fig. 2. 

The daia for the chilled amalgams are marked by dots and 
for the slowly cooled amalgams | by crosses. 

In the slowly cooled amalgams, as in the chilled, the rise 
of E.M.F. near B is at most very gradual. But, ‘owen C, 
the former amalgams behave differently. The rise is much 
greater than in the chilled amalgams, and the E.M.F. of a 
slowly cooled amalgam is now considerably greater than that 
of the corresponding chilled amalgam. 

This effect is exhibited numerically in Table A, and has 
already been alluded to in $6. It may be explained as 
follows :— 

We assume, for reasons given in § 5, that the solid grains 
in partially fluid chilled amalgams are of more uniform 
composition than those in similar slowly cooled amalgams. 
A chilled amalgam which is nearly all solid at any temper- 
ature @ will consist mainly of solid y per cent Cd amalgam, 
together with a small amount of fluid containing w per ce nt. 
Cd. 

A slowly cooled amalgam, of the same average composition, 
will contain a smaller proportion of solid grains ; but these 
wili be richer in Cd. Their surfaces will contain y per cent. 
Cd; but their interiors will contain more. If a sufficient 
number of these grains be present in the surface originally, 


as a Standard of Electromotive Force. 219 


or rise thereto after detachment by beat, they may produce 
a thin surface skin containing more than y per cent., as 


Fig. 4. 


seTestessdssenIEPETERETTEOCOETZ SSG? 


eee et 
EEE EEE EEE EEE eee 
oe 


jeueescise a 
RERSL JSS ROSNER Ieee 
Pir rrr Prityy 
Prey rt Pity prriyry 

pry prry ty 


seaenaecersesais 


ES ERRSRRREEA! 

SERRSER RES a! 

BESBERESSaSS 

trtttt —----—— ———-= 

Saas SeSEsEaaleeEeeeeeeeseeeeeeeeeeeesasseeEs 
Poe ee 


eer. oe =a = 
om GbacceseGenecoeaoaaee EEE EEE EE EEE eee 
BREE EERE Eee 
CO eee oa tet See eeen 
root 


described in §8 above. The slowly cooled amalgam will 
then give a greater E.M.F. than the corresponding chilled 
amalgam. 


220 Dr. S. W. J. Smith on the Weston Cell 


The excess over y per cent. Cd in the surface will only be 
appreciable when the surface skin is relatively thick, 2 e. 
when the surface grains are relatively numerous. For much 
of the under surface of the skin will be in contact with 
liquid w per cent. alloy, and cannot therefore contain more 
than y per cent. Cd. 

Thus it is only in the stronger slowly cooled amalgams 
that the E.M.F. can be much greater than that of the two- 
phase alloy—in agreement w ith fig. 4. 

In keeping with this view of the behaviour of the stronger 
slowly cooled amalgams, the percentage of Cd in the surface 
skin of any of them may increase within certain limits as 
the temperature rises. Thus when the temperature of one 
of these amalgams is raised from @ to @’, the solid grains 
must partially liquefy. By this means the percentage of Cd 
in the fluid alloy is raised from x to @', and the percentage 
of Cd in the new surfaces of the solid grains is y', greater 
than y. Electrolytic effects may ensue as before and cause 
the surface skin percentage to rise above y’. 

As the temperature is raised the surface grains may 
become relatively more numerous at first, since additional 
grains may float up to the surface; but in the end the 
surface skin will grow thinner and the E.M.F. will approach 
nearer to that of the two- phase amalgam corresponding with 
the temperature of observation. 


$10. Skin effects in all-solid amalgams.—It will be obvious 
that electrolytic skin effects of the kind described in § 8 

may occur in amalgams which are already all-solid, but in 
which ordinary diffusion effects are still proceeding with 
appreciable velocity. In such cases the surface film may 
acquire a larger share of the effects of diffusion than areas 
just below the surface, with the result that these may be less 
rich in Cid than the surface layer itself. 

Evidence of this condition of affairs is revealed when the 
data of Tables II. and VIII. are compared as in Table A 
above (see § 6), where it is shown that the surface begins to 
change at a temperature lower than that at which it would 
if no weaker amalgam were present near it. 

Similar considerations might explain otherwise puzzling 
time effects exhibited by the chilled amalgams of Table VI. 
(J. c. p. 266). In these, practically without exception, the 
marked increase of E.M.F. which occurs in the first few 
days is succeeded by a small but unmistakeable decline. An 
effect so general cannot be due to accidental irregularities 
in the process of crystallization. 


as a Standard of [lectromotive Force. 221 


The statement that “it may be due to the chilling pro- 
ducing a too highly concentrated amalgam in the outer 
shell” scarcely removes the need for further explanation. 
Effects of this kind are not confined to the chilled amalgams. 
The 15, 17, and 19 per cent. amalgams of Table IT. show 
appreciable decline in the surface percentage of Cd during 
the three months at 15° C. 

Another possible cause * of a slow decrease of E.M.F., 
which should be mentioned, depends upon the deduction 
(from § 12 below) that the equilibrium concentration of Hg 
salt in solution round an amalgam must be less than that 
round pure mercury. In cells of the Weston type, Hg salt 
must thus be diffusing continuously from cathode to anode 
and the amount of Hg in the latter must be slowly increasing 
by precipitation. Fortunately, the diffusion will generally 
be so slow that its effect upon the H.M.F. of a two-phase 
amalgam will remain unnoticeable for a very long time. 
But if the anode surface consists of a thin single-phase skin 
the percentage of Hg within it may increase perceptibly ina 
comparatively short period. 


§11. The question of the horizontality of BC in fig. 2.— 
‘The skin effects described in § 9 arise primarily trom the 
lightness of the solid grains and from their want of uniformity 
of composition. As soon as such effects become appreciable, 
departure from horizontality must ensue. A measure of the 
importance of these effects is given by the difference 
between the H.M.F.s of the richer amalgams, chilled and 
slowly cooled, of fig. 4. 

But another question remains. If the amalgams (chilled 
or otherwise) were of quite uniform composition between 
B and C, differing only in the relative amounts of 2 per 
cent. and of y per cent. amalgams present, would BC be 
absolutely horizontal, 7. e., would the E.M.F.s of all the 
amalgams within this region be found to be identical, how- 
ever refined the means of comparison might be ? 

The data of fig. 4 seem to establish the fact that there is 
always a slight risé from B towards C. In the region near 
B the amount of solid amalgam is so small that there is 
little room for appreciable variation in the percentage of Cd 
between the centres and surfaces of individual grains. 
Hence, as the electromotive data show, chilled and slowly 
cooled amalgams containing the same percentage of Cd must 


* Possible effects at the cathode are outside the range of the present 
paper. 


229 Dr. 8. W. J. Smith on the Weston Cell 


here be of practically identical composition. But even in 
this region the E.M.F’. curve slopes upwards. 

It is thus impossible to say beforehand that the E.M.F. does 
not rise as the percentage of Cd in the amalgam increases, 
even when the crystals are as uniform as they can possibly be. 


$12. The possibility of equilibrium between two amalgams 
and the same electrolyte.—The question raised in the pre- 
ceding section cannot be answered satisfactorily without 
more careful consideration of the conditions of equilibrium 
between each amalgam and the electrolyte than has been 
so far necessary. 

It will perhaps be useful to indicate first how, neglecting 
surface energy, the conditions of equilibrium between the 
two homogeneous amalgams can be represented thermo. 
dynamically *. 

The total energy e of « homogeneous substance containing 
masses m, and m, of its two components can change by 
acquisition of heat (alteration of the entropy 7), performance 
of external work (alteration of the volume v) and change of 
composition (alteration of the mass of either constituent). 

For a reversible change we may write 


_ (0€ & fel ) O€ 
vi ei (Sein 2” a She. i (ea YUVM5 otis +(e ii 


The values of the first and second partial differential 
coefficients are obviously @ and —p. The terms containing 
them represent the energy variation due to change in the 
heat content and volume of the working substance; the 
remaining terms indicate how the energy variation depends 
upon the composition. The partial differential coefficients 
which they contain are functions of the composition of the 
working substance and we may write 


de = Odn—pdv+ py dmy+t py dm. 


For any other homogeneous mixture of the same sub- 


stances, also capable of reversible variation, we may write 
de! = Ody! —p dv’ + py! dmy' + ps’ ding. 
Now suppose that these two mixtures can coexist in 
equilibrium. 
By hypothesis the components are independent variables. 


We may imagine that a small quantity dm of the mm com- 
ponent leaves the second phase and enters the first in such a 


* Cf. Gibbs, Trans. Conn. Acad. vol. iii. pt. 1, p. 115. 


5 
Dae 


as a Standard of Electromotive Force. 223 


way that the transference takes place without change in the 
entropy or volume of either phase. The energy variation of 
the system would then be 


/ 
€ € . s 

= | dm, + (<<) dm,! = (f4,—4y')dm. 

om, 7VMy Om, y'v'm'g 


But since neither the entropies nor the volumes change, 
the system neither does external work nor acquires heae 
Consequently we must have 


rae 
by—py = 90, 
and, by a similar argument, 


[la — My! = 0). 


These and two other equations, derived one from each 
phase at given 6 and p, suthce to define the conditions of 
equilibrium completely—the essential variables being 


m/v, m/v and m/v’, m'/v' respectively * 


In assuming, as above, that the total energy of a known 
mass of each phase, at given @ and p, is dependent only on 
its composition, we neglect the possible influence of surface 
energy. 

We may continue to suppose that, to a first degree of 
approximation, the surface tension and electrostatic potential 
ditference between the « per cent. and y per cent. amalgams 
are negligible and proceed to examine the conditions under 
which the two amalgams could coexist in presence of the 
same electrolyte. 

Considering first the equilibrium of fluid amalgam and 
electrolyte, reversible exchange of Cd or Hg between elec- 
trode and electrolyte will be possible since the latter is a 
solution of sulphates of Hg and Cd. But while the masses 
dm, and dm, of the metals may be considered neutral in the 
amalgam, they are associated with definite positive charges, 
which we may write Ay dm, and k,dmg,, when in solution. 

In order that (as in the equations already given) we may 
still regard m, and m, as independent variables, we assume 
that when a quantity dm, of mercury leaves ihe solution, 
a quantity dm3 of anions (SO,) carrying the charge 


ks dms = —h, dm, 


also leaves the solution and accumulates at the surface 
separating electrode and electrolyte. We assume also that 


* Of. § 14, below. 


224 S. W. J. Smith on the Weston Cell 


a similar effect cpanel the transference of dm, of 
cadmium. 

In consequence, jf we suppose the electric potential of the 
amalgam to be V and of the electrolyte to be V’’, it wiil be 
seen that the. reversible energy variation may now be 
written 


de + de!’ = 0(dn+dn") —p(dv + do") + Guy — pe, ")ding 
+ (peo pe! dmg + (V —V") (ky, dm, + ky dmg) 


+ (p13 — bs"") (7 dm,+ dm), 


in which Oc eee ra 
( Ta == pele ky V+ ets 
One, yvmMe U3 


with similar expressions for the other quantities, 4; referring 


to the surface layer and yu,’ to the interior of the electrolyte. 
For equilibrium, as before, 


i I | 
(44 — py") +A(V —V") + 7 ers) = 0, 


and (po— py )+h(V— Lae (a js )=0 


ae eae 


In the same way, considering the wh amaloeam and the 
electrolyte, we must have 


(Hy! — an") + x(V'—V") + Fs’ Hs”) = 0 | 
and ' 
| 


(Hy! — pla”) + ko (V! La eee Bs —pys) = 0. 


hy 


J 


Considering the equilibrium of ins amalgams with each 
other we assume 


fa = ee 2 = fy! ° and V = Ve 


Suppose now that we ignore possible differences at the 
surfaces separating the electrolyte and the fluid and solid 
amalgams respectively and put 3 = j43° (see also § li 3, below). 

Then it will be possible to find values of py", po’ ' and fs, 
which will be related in such a way as to satisfy the above 
equations simultaneously and make 


VV" = Voy 


In other words it will be possible to find electrolytes 
within which the two amalgams can exist side by side in 


as a Standard of Electromotive Force. 225 


complete equilibrium. With any one of these electrolytes 
the E.M.F’. would be independent of the relative amounts of 
the two phases present. 


§ 13. The effect of surface energy.—lf, as above, the surface 
energy variations of the liquid and solid amalgams are 
regarded as identical, the conditions of equilibrium assume a 
simplicity which there is no doubt they do not possess. 

We do not know the relative importance of the energy 
per unit area of the surface separating the amalgams ; but 
the difference between the energies of the surfaces separating 
the respective amalgams and the electrolyte is perceptible. 
The liquid amalgam appears always to spread over the 
surface of the solid amalgam in the presence of the electro- 
lyte. We may therefore assume that the surface energy 
between solid amalgam and electrolyte is greater than that 
between liquid amalgam and electrolyte. 

There are phenomena which show that the relation between 
surface energy and potential difference is often very complex 
and § 12 is put forward merely as a method of representing 
the general nature of what occurs. It can be seen, however, 
that even if the conditions of equilibrium were as simple 
as those already given, the difference of surface energy 
would cause 43; to exceed yz. The two amalgams could not 
then, subject to the condition V—V'" = V’—V”, be in 
equilibrium with the same electrolyte. 

The value of V’’— V' satisfying the second pair of equations 
would be greater than that of V"—V satisfying the first two, 
2. e., the potential of a given electrolyte with respect to the 
solid amalgam would exceed that of the same electrolyte 
with respect to the fluid amalgam. Hence, supposing 
V=V’, the amalgams could not coexist in contact with the 
_ same electrolyte. 

Such considerations are enough to show that there is no 
theoretical necessity for horizontality of BC. It is much 
more likely that the equilibrium at the anode is between a 
variable phase and the electrolyte, and that the E.M.F. electro- 
lyte/electrode is distinctly greater when the amalgam is one 
which is just on the point of partial liquefaction at 6 than 
when it is one in which the last traces of solid have just 
disappeared. ; 

There is therefore every probability * that the E.M.F. of a 
cell of the Weston type must always be to some extent 
dependent upon the percentage of Cd in the amalgam. 

* Mr. F. E. Smith has kindly supplied me with further details of some 
of the measurements recorded in his Table VIII. which confirm this 
view. 


Phil. Mag. 8. 6. Vol. 20. No. 115. July 1910. Q 


226 Dr. S. W. J. Smith on the Weston Cell 


§ 14. The application of the phase rule to cxdmium amal- 
gams.—The considerations given at the beginning of § 12 
were introduced in order to explain a method of regarding 
the effects of surface energy. They also supply a means of 
interpreting the “equilibrium curves” of fig.1. As in § 12, 
if two homogeneous phases containing components A and 
B can coexist in equilibrium we must have not only equality 
of temperature and pressure, but also two other equalities 
which we may write fa=/a and pe=p!. 

I'o determine completely, at given @ and p, the state of 
any mixture of A and B, we require to know how these 
‘potentials’? vary with c, the percentage of B in the 
mixture. 

It is conceivable that the substances can form a series of 
mixtures in all proportions and that any one of these can he 
entirely fluid or entirely solid at the temperature @. Tor a 
homogeneous mixture, at constant temperature and pressure, 
we must have 


Mad a + mol po =0. 


Considering all-fluid mixtures first we may suppose, 
following Gibbs *, that, when ¢ approximates to zero, pz must 
have a very large negative value, whilst mw, is finite and 
duz/de has a finite negative value. Similarly, when ¢ 
approximates to 100, u, must have a very large negative 
value, while yz is finite and dy,/de has a finite positive 
value. 

The curves connecting the variations of uz and py with c 
may no doubt be complicated ; but, in the simplest cases, 
they may (from what precedes) take forms like AM and BN 
of fig. 5. 

We may assume that analogous relations, represented by 
the curves A’M’ and B'N’, hold for the all-solid mixtures. 

If, as in fig. 5, a temperature is chosen which is below 
the freezing point of B, but above the freezing point of A, 
we shall have u,>,,' for the phases of pure B, and hence B 
will lie above B’ ; but a’> py, for the phases containing pure 
A, so that A’ will be above A. 

The conditions for coexistence of a fluid phase containing 
x per cent. B and a solid phase containing y per cent. B are 


[Ma le=x = [ Ma’ Joxy 
and (pb lea = | eae 


cide Oy cine td tS 2 


as a Standard of J¢lectromotive Force. 227 
Inspection of fig. 5 will suffice to show the possibility of 
finding values of 2 and y which satisfy these conditions. 


Fig. 5. 
B 


The same conclusion can be reached, less directly, by 
considering the variations with ¢ of the total thermo- 
dynamical potentials € and ¢’ per 100 grams of all-fluid and 
all-solid mixtures, remembering the relations 


dé/de = pa—pfa and dé//de = p,'—p,/. 


In the system of fig. 5, therefore, coexistence of two phases 
is possible when the liquid phase contains per cent. B and 
the solid phase y per cent. B. All mixtures containing between 
x per cent. and y per cent. of B, and only those, can split into 
two phases. 

Q 2 


228 Weston Cell as a Standard of Electromotive Force. 


If, however, the temperature is below the freezing point 
of A, as well as below that of B, the relative positions of 
AM and A’M! will be reversed and it will now be impossible 
to find values of «# and y for which pz, wa and pe, fe 
respectively, are equal. In other words, one-phase solid 
mixtures only will be stable * 

If it is assumed that the ie variations for the mixtures 
of Hg and Cd used in Weston cells are of the form repre- 
sented (diagrammatically) in fig. 5, it can at once be seen 
why the amalgams exhibit the properties which have been 
discussed. 


§ 15. The temperature coefficients of cadmium-mercury 
cells—The efficiency of the Weston cell as a standard does 
not depend only on the fact that the chemical composition of 
the amalgam can vary within considerable limits without 
producing more than a few millionths of a volt difference in 
the electromotive force ; but also upon the extreme smallness 
of the effect of temperature change near 0° C. 

It is instructive to consider how the existence of the two- 
phase amalgams happens to be the cause of the second 
phenomenon. 

The data of Table XI. (/. ¢. p. 273) show that the smallness 
of the temperature coefficient is not due to absence of heat 
exchanges with the surroundings during isothermal working 
of a two-phase cell, but to the fact that these nearly balance 

each other. Somewhere near 5° ©. there is an exact balance 
and the temperature coefficient vanishes. 

Above 5° C., the temperature coefficient has a eal 
negative value. This means that slightly more heat escapes 
from the cell (during action) than it absorbs from the outside. 
The presence of the two-phase amalgam reduces the net loss 
of heat. For, during the working ‘of the cell, the average 
percentage of Cd in the anode must diminish and the 
equilibrium compositions of the coexisting phases can only 
be maintained by reduction in the amount of the richer 
(solid) phase. This will occasion absorption of heat and so 
reduce the net amount evolved. 

Below 5° C., the absorption just mentioned slightly 
overbalances the remaining effects and the temperature 
coefficient is positive. 


* Cf. § 5 above. 


st rane aa J 


XVI. On the Shape of the Atom. By R. D. Kieemay, 
D.Sc., B.A., Mackinnon Student of the Royal Society ; 
mmanuel College, Cambridge”. 


T the absolute zero of temperature the molecules of a 
substance would probably be in contact with one 
another. From a knowledge of the density of different 
substances at the absolute zero we could therefore determine 
the real relative volumes of different atoms and molecules. 
The density of a substance at the absolute zero cannot be 
found directly, but it can be calculated with probably fair 
accuracy. Such a calculation has been carried out by 
Guldberg ft for a number of substances. And Traubet has 
shown, using these determinations, that the volume of an 
atom is proportional to the square root of its atomic weight, 
and the volume of a molecule therefore proportional to the 
sum of the square roots of the atomic weights of the atoms 
composing the molecule. A knowledge of the connexion 
between the volume of an atom and its atomic weight does 
not by itself furnish any information as to its shape, but this 
relation in conjunction with the cross-section of the atom, 
which can be obtained from the kinetic theory of gases, gives 
us some information on this point, as will be shown in this 
paper. 

The shape of the atom which suggests itself as the most 
probable, and which is the one usually assumed, is that of 
the sphere. Assuming then that the atom is spherical in 
shape, we have that its volume is proportional to 7° and its 
cross-section proportional to 7?, where ” is the radius of the 
atom. Since its volume is also prcportional to m!/?, where 
m is its atomic weight, its cross-section is proportional to m*, 

In Tables I., II., and III. values of Q, the sum of the 
diametrical sections of the spheres of action of the molecules 
contained in unit volume of a gas at atmospheric pressure, 
are given for a number of vapours. The values contained in 
Tables I. and II. were taken from Meyer’s ‘ Kinetic Theory 
of Gases,’ pages 303, 307, and 308, and those in Table III. 
were obtained from Landolt and Bornstein’s Tables, 5th 
edition. They correspond to a temperature of 0° C. The 
sum of the sections Q is obtained from the equation 

1 
oe end V8 1, 
where s is the mean radius of the sphere of action of a 
* Communicated by the Author, __ - . 
+ Zeit. fiir Phys. Chemie, xxxii. p. 122 (1900). 
t Phys. Zeit. p..667, Oct. 1909. ome 


230 Dr. R. D. Kleeman on 


molecule, N is the number of molecules in unit volume 
at atmospheric pressure, and L is the mean free path of a 
molecule. Since N has the same value per unit volume for 
all gases at the same temperature and pressure, the diametrical 
sections of any two molecules are to one another as the cor- 
responding values of Q. The tables contain also the values 
of 2m, the values of m used for their calculation being 
given at the head of Table I. The last column of each table 
gives the ratio of Q to Sm. 


TaBLeE I, 
Cube roots of the atomic weights of a number of atoms: 


H=1, C=2:29, N=2-41, O=2°52, Fl=2°67, S=3°18, 
C1=3:29, Br=4-31, [=5:03, Hg= 5°85. 


Q 
1/3 em Sa 
Gas Q. am : Sei? 
Beet re neki sees. 18,700 4°81 3,888 
ok) | SEE ee 27,000 7°33 3,684 
Bs sabia duciiwne 18,600 4°82 3,859 
(00 es ee 27,100 7°34 3,693 
1 19,200 4:93 3,895 
See 17,400 5°04 3,452 
EES hibvniacessns 22,200 6 29 3,530 
BREE Wescewsrwses 43,500 13°10 3,321 
REPO, Soitececmee- 49,300 12°87 3,830 
EE Ge ee 9,900 2:00 4,950 
Ec icwdecaue oxer 24,900 5°41 4,603 
BE din cevdesnses 42,500 8:58 4,953 
CL 0 a ae 40,100 8:58 4,673 
BM he teats deen we 43,900 9:42 4,660 
SOLS ee me 37,900 8:22 4,610 
MIMO dase olde ves 25,100 4°29 5,851 
SS ee 38,800 6°58 5,896 
TOON Goa sesicoce sa. 24,900 4°52 5,509 
[EV Ck eee 29,300 518 5,657 


If the atoms of a molecule all lie in a plane, then, as Meyer 
has shown in his ‘ Kinetic Theory of Gases,’ pp. 304-309, 
the cross-section of a molecule ought to be approximately 
equal to the sum of the cross-sections of the component atoms. 
The cross-section of a molecule should therefore be an additive 
quantity relating to its atoms, and this Meyer finds to be, 
approximately the case. If, further, the shape of the atom 
is spherical and its volume proportional to m1, the cross- 
section of a molecule will be proportional to $m, and the 
ratio se will be constant. This, as will be seen from the 
tables, is to a certain extent realized. 


| 


the Shape of the Atom. 231 
TaBLE II, 
Alcohols. 
; Q 
pa Sontt snl? 
Methyl Alcohol... CH,O | 49,000 — 8:81 4,418 
Etliyl ee gee Vi Os CO, 64,790 13°10 4.939 
Propyl ie a tay CLO 87,100 17°39 5,009 
Butyl ey OM Lan 107,800 21°68 4,973 
hksobatyl 5.0 -dy OH, O VE AGO) ecco Wrn. fads en's 4,852 
| Amyl il nee, 0) 127,000 25°97 4,890 
| Hexyl nee EL ©) 159,300 30:26 5,144 
Acids. 
Formic acid ...... CH,O, 43,900 9°33 4,706 
PARORRIA NS vay) cadets C,H.0, 59,500 13°62 4,359 
Pr opionic acid 27.011 .O), 77,900 17°91 4,349 
Butyric __,, Be tO), 106,500 22°20 4,797 
Isovaleriani: acid. C5H,,0,| 142,600 26°49 5,395 
Ethers. 

. 89,700 4,149 | 
Mer ie SOON on aan | 21-62 3716 | 
Methy] ether ...... C,H,O 43,500 13:10 3,321 

Kisters. 
Methyl formate... C,H,0, 56,700 13°62 4,164 
Methyl acetate ... C,H,O, | 78,900 17-91 4,406 
Ethyl formate ... 83,500 5 4,662 
Ethyl acetate ... C4H,0, 102,000 22 20 4,594 
Methyl propionate __,, 92,600 By ona) 
Propyl tormate : 98,900 ms 4,454 
116,300 ) : 4,390 
Ethyl] propionate C, ait ber 87, 500 } 26 49 3,303 
Isobutyl formate " 86,700 “5 3,273 
Methyl! butyrate 34 115,500 My 4,360 
, 111,200 4,198 
Methyl isobutyrate a 88,400 } ” 3537 
Propyl acetate ... ,, 90,700 %9 3,424 
Ethyl butyrate... C,H,,0,| 129,300 30°78 4,200 
Ethyl isobutyrate a3 123,000 a 3,996 
Isobutyl acetate fi 133,600 Me 4,341 
Propy! propionate % 136,300 i 4,429 
Ethyl! valerianate C,H,,O, 149,000 30°07 4,248 
Isobutyl propionate __,, 152,400 5 4,344 
Propyl butyrate - 145,600 #3 4,142 
Propyl isobutyrate ,, 137.700 4g 3,926 | 
Tsobutyl butyrate ©,7,,0,, 165.700 39°36 4,209 | 
Tsobutyl isobutyrate _,, 165,200 - 4,197 [ 
Propyl valerianate ,, | 163,700 A 4,159 | 
Amyl propionate 177,100 - 4,499 
Amy] isobutyrate C, TH Oy | 185,600 43°65 4.246 
Isobutylvalerianate ,, | 186,500 % 4,273 | 


232 Dr. R. D. Kleeman on 


The deviations from constancy are more likely due to the 
values of Q not giving the true relative values of the sections 
of the molecules, than to the atoms not being spherical in 
shape, or their volumes not being proportional to m'”. The 
value of Q must obviously include a certain amount of space 
not occupied by the atoms, especially that situated near the 
centre of the molecules. The amount of included space in a 
molecule wiJl depend on the arrangement of its atoms. 
Probably the variation of Q with temperature is largely due 
to a variation of the amount of unoccupied space included 


in Q. 

The values of oe in Table I. show a distinet tendency 
to fall into three groups, the values of each group being 
approximately constant. Each group probably corresponds 
to a certain arrangement of the atoms corresponding to which 
a certain proportion of external space is included in Q. 

The same tendency is exhibited by the values in Table II. 
The values belonging to the same chemical group are approxi- 
mately constant. This is best shown by the alcohols and 
esters. In a few cases two values of Q for the same vapour 
are given in the table corresponding to different observers. 
The large differences that sometimes exist between the results 
of different observers, which are probably due both to errors 
of experiment and different methods of measurement, suggests 


that the deviations from constancy of the ratio =—, are to 
7 Smile 


a certain extent due to erroneous values of Q. 


TaBsueE III. 


| ed) 

Q. Mane Sit 2 
OGG. 2230-5. 68,000 17:20 3,954 
OsHE@I Br... 63,000 16:18 3,894 
led: See 74,000 16:22 4,562 
Ch 49,900 14°61 8,415 
(Coa ee 74,600 23°19 3,224 
EIS ons «sie: 42,300 10°32 4,100 
BE Bs aap 51,400 585 8,788 
LE, URE ae eee 11:70 4,394 


Table III. contains the ratio of = for a few molecules 
which contain heavy atoms. It will be seen from the table 
that if the mercury molecule is taken as monatomic the value 


the Shape of the Atom. 233 


of the ratio is much larger than it should be on that assump- 
tion, but if it is taken as diatomic the ratio fits in well with 
the others. This suggests that a molecule of mercury in the 
gaseous state at zero temperature is probably diatomic. 

We have compared the values of @ for different gases at 
zero temperature. It is probable, however, that the proper 
temperatures for comparison are corresponding temperatures. 
The range of the data is, however, not sufficiently extensive 
to allow a comparison to be made for corresponding states. 

The coefficient of viscosity of a gas is given by the 

mv 
Aas?” 
a molecule. The equations for 7 and Q thus both contain 
the square of s, and the relative values of each of these two 
quantities for the substances belonging to one of the groups 
given in Tables I. and II., can therefore be obtained with the 
same degree of accuracy from these equations on m/* being 
substituted for s’. This applies also to the coefficient of 
diffusion 6 of a molecule in a gas composed of molecules of 


3 
the same kind, which is given by 6= als where p denotes 
the pressure of the gas. 12 pas’ 

The coefficient of diffusion 6,, of a gas 1 into a gas 2 is 
given by the equation 


O12 


equation 7= where v is the velocity of translation of 


K 
1/2 


— Mos (sy +82)? ? 


where K is constant at constant pressure and temperature, 
and s,, Ss, are respectively the radii of the spheres of action 
of the molecules 1 and 2. Now s, and s2 are, according to 
the results obtained in this paper, proportional to (2my°)"” 
and (Xm3'*)’” respectively, and the above equation may 
therefore be written 


aaa 10 ISIE al a 
12-= ms” Sai”)? ae (Sm) 1/2 } 29 


where K, is a constant. This equation may be used to 
calculate relative vaiues of 8)». 

Table IV. contains the coefficients of diffusion of a number 
of gases into each other, they were taken from Meyer’s 
‘Kinetic Theory of Gases, p. 275. The values in the table 
labelled “ calculated’ were obtained by means of the above 
formula. The value for N,O—CO, was put equal to the 
observed and the other values reduced correspondingly. 
The calculated values, it will be seen, are approximately 


234 Dr. R. D. Kleeman on 


equal to the observed; the agreement, however, is not very 
good. The equation may thus be used to obtain a rough 
idea of the value of the coefficient of diffusion of two gases 
into one another. 


TABLE LV. 
+ | 
Coefficients of diffusion. 

| Diffusion of gas Calculated by 

takes place from Observed. Calculated. another 
right to left. | formula, 

HY OG. cectsn ons "722 948 ‘S71 
Ee OO ow: 642 ‘940 “904 
By CO, oa 556 “724 723 
BE SO aes nsto ‘480 ‘668 ‘655 
) Li 0 ai ane ‘180 "166 "116 
SS 0 Spo ‘161 °125 ‘160 
OOl— COs 2... 6. “160 136 *125 
CH,—CO,......... "159 "159 ‘151 
Air —CO.,,......... 142 | ‘136 "125 
N,O—CO,,......... ‘089 ‘089 ‘089 


The formula for the coefficient of diffusion used in this 
paper is the one usually given in treatises on the kinetic 
theory of gases. Maxwell has given another formula for 
the coefficient of diffusion based on the assumption that the 
force of attraction or repulsion between two molecules varies 
inversely as the fifth power of their distances of separation. 
This law, the writer* has shown, holds approximately for 
distances of the order of the distances of separation of the 
molecules in a liquid, and the attraction is further propor- 
tional to the product (Zmj”)(Smz”). Maxwell’s equation 
accordingly becomes 


ee M1 + Mg 1/2 
eee es one =m } , 


where K, is constant at constant temperature and pressure. 
The values of 6). obtained by this equation are given in the 
fourth column of Table IV. The values of 6, calculated by 
these two different formule agree better with one another 
than with the observed values. On the whole both agree 
equally well with the facts. 


* Phil. Mag, May 1910, p. 783. 


the Shape of the Atom. 239 


Some further deductions of interest can be made relating 
to the properties of the atom. If the mass of an atom is 
denoted by m, its volume, we have seen, is proportional to 


m?,and its density therefore proportional to an or m}?, 
Thus the density of atoms increases with increase of atomic 
weight. The density of a lead atom is thus about 14 tinies 
that of a hydrogen atom, and the density of a hydrogen 
atom about 30 times that of an electron. That the density 
of an atom should increase with the atomic weight we should 
expect since there would be a tendency of the atom to contract 
under the mutual attraction of its parts, and this would 
increase with the mass of the atom. 

The writer* has shown that the attraction other than 
gravitational between two molecules a given distance apart 
is proportional to the product (mj")(2m3"). This attraction 
gives rise to the surface-tension of liquids, chemical combi- 
nation, &c. This fact may now be stated in a different form, 
namely, that the attraction is proportional to the product of 
the volumes of the molecules. It was also shown in the 
paper mentioned that the attraction of several atoms close 
together, as they occur in a molecule, is not exactly 
additive. Wesee now how this may be caused. Whena 
number of atoms concentrate to form a molecule each atom 
must contract slightly owing to the attraction of the different 
parts of the molecule on one another. The attractive 
force of each atom at an external point is therefore less 
than it would be if the other atoms were absent. The con- 
traction of the atoms would be greatest at their surfaces of 
contact, which would have the effect of increasing the extent 
of the total surface of contact. The stability of the structure 
of the molecule would thereby be increased. 

An electron in the neighbourhood of an atom will be 
attracted by it due to electrostatic induction. If the atom is 
considered a perfect conductor of electricity the attraction 


is given} by 
"ead (2f? a’), 
fip—ey 
where a is the radius of the atom, and 7 the distance of the 
electron from the centre of the atom. When /f is large in 


* Loc. cit. 
+ Maxwell’s ‘ Electricity and Magnetism,’ vol. i. 3rd edition, p, 251. 


236 Dr. R. D. Kleeman on 


2,3 
: ; ‘ ea 
comparison with @ the expression becomes 


Now a? is 


ids) e 


proportional to the volume of the atom, and this, we have 
seen, is proportional to m¥?, and the attraction is thus pro- 
portional to the square root of the atomic weight of the 
atom and inversely proportional to the fifth power of the, 
distance of separation of the electron from the centre of the 
atom. It is of interest that the ‘ chemical” attraction 
between two atoms follows a similar law. Thus the writer 
has shown in the paper mentioned above that the “chemical” 
attraction between two atoms is proportional to the product 
of the square roots of their atomic weights, or, if one atom 
is always the same, proportional to the square root of the 
other atom, and inversely proportional to the fifth power of 
their distance of separation. The above result is of interest 
and importance in connexion with the passage of @ or 8 
particles through matter. 

It is also interesting to note that the forces are of the same 
order of magnitude. Thus if we substitute 3°4x10-! for e 
and 10-* for a, the expression for the electric attraction 

.F ~44 
becomes = ise . The constant K relating to the 


0 


chemical attraction between two atoms—say of lead, corre- 
sponding to the above constant (e?a*), was calculated to be 
equal to 4:14 x 10-“, which is of the same order of magnitude 
as the above value. 

When a 8 particle in passing through matter encounters 
an atom it gets deflected from its course and also produces 
secondary @ rays from the atom. The amount of secondary 
radiation, and its direction of propagation and that of the 
primary 8 ray after an encounter, will depend on the nature 
of the encounter. Let us suppose the secondary radiations 
from a large number of atoms taken at random are made 
from the same atom with the paths of the primary rays 
parallel to one another, the deflected primary rays being 
included in the secondary radiation. The relative distribution 
of the secondary radiation round the atom in direction of 
motion with respect to the direction of motion of the primary 
electrons, and the distribution of velocity among the electrons 
moving in any given direction, will be independent of the 
number of atoms considered if a sufficiently large number is 
taken. The various angles made by the secondary rays with 
the direction of propagation of the primary 8 rays will be 
grouped about a mean somewhat like the molecular velocities 
according to Maxwell’s law. The value of this mean angle, 


the Shape of the Atom. 256 


or the distribution of the radiation, must depend to a certain 
extent on the nature of the atom. The distribution of the 
secondary radiation outside an atom produced bya # ray 
must obviously depend largely on the amount of matter of 
the atom traversed by the 8 ray. Therefore, if the atom is 
spherical in shape a given increase in the atomic weight 
would produce less change in the magnitude of the mean 
angle of distribution when the atom is large than when it is 
small, since the change in the diameter of the atom will be 
smaller in the former case than in the latter. We should 
therefore expect the limiting distribution of the secondary 
radiation to depend less on the nature of the atom the greater 
its atomic weight. 

This could be tested by means of the following arrange- 
ment. Suppose we have a number of slabs of different 
kinds of matter of infinite thickness, and some of the atoms 
in each slab eject electrons at right angles to one of the faces 
of the slab with the same velocity, which is also independent 
of the nature of the matter. The radiation from both of the 
faces of a slab would consist of real secondary radiation and 
deflected primary rays. Now when we are dealing with 
slabs of infinite thickness the ratio of the two radiations from 
the opposite sides of a slab is independent of the density of 
the slab; it is also independent of the proportion of the 
atoms which eject 8 rays. These ratios for slabs of different 
substances would therefore afford some information on the 
relative limiting distribution of the secondary radiation round 
the atom. 

Now the experiments of Prof. Bragg and Dr. Madsen* on 
the secondary cathode rays ejected by y rays satisfy the 
above conditions. They showed that the electrons ejected 
by y rays move in the direction of propagation of the rays, 
and if the rays are hardened by being tirst passed through a 
thick screen of lead, the velocity of these electrons is practi- 
cally independent of the nature of the matter in which they 
are produced. They next carried out a set of measurements 
of the amounts of secondary cathode radiation from the two 
sides of plates of different materials sufficiently thick to give 
the maximum amount of radiation —which is equivalent to 
dealing with plates of infinite thickness, the y rays passing 
through each plate at right angles to one of the surfaces. The 
ratios of the radiations from the two opposite sides of each 
plate for hard y rays have been calculated from a table given 


* Phil. Mag. xvi. pp. 918-99, Dec. 1908, 


238 Sir J. J. Thomson on the 


in the paper mentioned above, and are plotted in the diagram. 
It will be seen that the ratio changes very rapidly with the 


ood 


FPA T/0 OF SECONDARY RADIATIONS 


Aromtc W7 —— > 


constant for atoms of high atomic weight. The relative dis- 

tribution of the secondary radiation round an atom thus 

gradually becomes constant as its atomic weight increases, 

and this we have seen is most likely to be the case if the 

atom is spherical or approximately spherical in shape. 
Cambridge, April 23, 1910. 

XVII. On the Theory of Radiation. By Sir J. J. THomson, 
M.A., F.R.S., Cavendish Professor of Experimental Physics, 
Cambridge *. 

We the Philosophical Magazine for August 1907, I discussed 

a theory of radiation from hot bodies which regarded the 
radiation as arising from the impact of negatively charged 
corpuscles with the molecules of the body; the impact starting 

* Communicated by the Author. 


Theory of Radiation. 239 


electric pulses which collectively constitute the radiation from 
the body. When we resolve, by Fourier’s theorem, this 
radiation into its constituent harmonic vibrations, we find that 
the amount of light of any given period depends upon the 
ratio of that period to the time occupied by a collision. It 
was shown, moreover, that this radiation would not conform 
to the Second Law of Thermodynamics unless the time 
occupied by a collision varied inversely as the kinetic energy 
of the corpuscle before it came into collision, and in addition, 
that the time of collision of a corpuscle moving with a given 
speed must be constant and independent of the nature of the 
molecule against which the corpuscle collides. I showed 
that the first of these conditions would be satisfied if the 
forces exerted during the collision between a.corpuscle and 
a molecule varied inversely as the cube of the distance 
between them ; the second condition will be satisfied if the 
collision is regarded as taking place, not between the 
corpuscle and the molecule us a whole, but as between 
the corpuscle and systems dispersed through the molecules, 
these systems being of the same character in whatever 
molecules they may be found, and repelling the corpuscle 
with forces varying inversely as the cube of the distance 
between them. Forces of this type would be exerted by 
electric-doublets of constant moment with their negative 
ends pointing to the corpuscles. 

In this paper I shall consider more in detail the collision 
theory of radiation when the forces exerted during collision 
vary inversely as the cube of the distance between the 
colliding bodies. In the paper already quoted it is shown 
(see Phil. Mag. xiv. p. 225) that if Hg be the energy per 
unit volume of the radiant energy with frequencies between 
g and g+dq, 

1 mK 
Eq= 3 ays 740, 


where m is the mass of a corpuscle, V the velocity of light 
wn vacuo, K the specific inductive capacity of the luminous 


body, and 


re fs Foblonsgh dX. 


where f(A) is the acceleration of the corpuscle at the 
time 2X. 

We shall first find /(X) when the repulsion varies inversely 
as the cube of the distance. If 2 be the distance of the 


240 Sir J. J. Thomson on the 
corpuscle from the system with which it collides, 


Been ahh 
ie 0. ‘“ : 


! - a. ee : 
gm oo 5 i vy, 


where v is the ee of the ee before the collision. 
If we take ¢=0 when the corpuscle is closest to the molecule, 
we get by integrating this equation 


eee he ¢? + aos 5 
MV 


f(t) the acceleration of the corpuscle is equal to #/mx’, and 
thus 


m 


pe 


Mv  mei(e + BV 
Mv 


= Ns b(t? cos qt dt 


mv? | —@ 9 B ad 
oe mise 
( Mv 


And 


To calculate the integral, let 


T° cos gt dt 
uU= 
ees aes C2) ay 


By differentiation, and aan by parts, we easily find 


ldu_ du : 
‘aya ae 
we tcg—2, 
ldu 2a 
ede dx 


If w= ww, this equation becomes 


Pag. + dean 1 
dv? +27, ~¥( 1+ 3) =0 ° . fs, as (1) 
Now the solution of the equation 


Gy) Wa. n? 
Age = —y(1+ 3%) =0 


xv dx 


is (since this is Bessel’s equation with cx for the independent 
variable) 


AL («) aE BK, (2) 2 


Theory of Radiation. ait 


where A and B are arbitrary constants, and 
=) (ia); 


es ae |, cos Ceaibh 6) 
: Jo cosh™ 


Tables for K, and K,, and I, and J, are given by 
Mr. W.S8. Aldis, Proceedings of the Royal Society, vol. lxiv. 
p- 203. 

]_(z) becomes infinite when 2 is infinite, while K,(z) 
vanishes in that case. 

We see that the solution of (1) is 
Since u and therefore w vanish when z is infinite, A=0, and 
we have 


dd*. 


u=«vw=BxeK,(2), 


shire Kx 1{ * cos (a sinh d) 
ie, Oe fied 


wv '9 cosh” p ad. 


Since when 2=0, 


d i 2 
ies eK,=— ( Spo boty 
9 cosh? 
we see that oes a 
Fe 
hence jock con ghdt 
b= a 


3 4 

TAD Ni jh \2 
po Wea cline 
MD 


=—2a/ 9K, (9 res , 
As }mv’ is the kinetic energy of a corpuscle, we have if 
the corpuscles are in thermal equilibrium with the body 
dmv’? =a, 
where @ is the absolute temperature of the body, and 
| a= 142 x 10-%, 
hence, if h=4\/ym, we have 
hg h 
i= —20( 73) (3): 
* See Gray and Matthews, Bessel’s Functions, p. 67. 
Phil. Mag. 8. 6. Vol. 20. No. 115, July 1910. R 


242 Sir J. J. Thomson on the 


and E, the energy of the radiation with frequencies between 


g and g+dq is 
h h 
ane aaa! erika, Gola qq. 


Denoting for evity the function «K,(x) by T(v) we may 
write this as 


8 a6 
3 ays l (hg/48)¢°dg. 


If X is the wave-length corresponding to the frequency q, 
“2a 
J mv VV; 
and KE, the energy in the radiation with wave-lengths 


between A and A+d\” will be 


8 ed am 
Pea (hq/ 


The value of q soe makes the ai of dd in this 
expression a maximum is evidently such that if “ = 2, 
then x is determined by the condition that #*®K,(x) should 
be a maximum; from the values of K,() given by Mr. Aldis 
in the Tables already referred to I find that a=2°4; hence 
if A, is the value of X which makes the energy beiwaas r and 
A+dX a maximum at the temperature 8 when dd is given, 
we have 


h2ar 2:4 

nab. ane 
or 4 g=27 V 
ia ha 2-4 


Thus 2,,0 is a constant, and this constant is known with 
consider aie accuracy fut the experiments which have been 
made on the radiation from a black body. ‘The value of this 
constant found by Lummer and Pringsheim * is 2940 x 10-4, 
If we call this quantity ¢, we have 


B= 5 San PQsgnO@r.. . . . @) 
Since T(x)=2K,(7)=—1 when «x is very small, we see 


that for long waves 
| Hi, varies as 3 
The same law is given by both Rayleigh’s and Planck’s 


values for E,. 
* Verh, deutscn. phys. Ges. 1. p. 230, 


Theory of Radiation. YA43 


Pat be ] 
Whien « is very large, then (Gray and Matthews, Bessel’s 
Functions, p. 68) 
Te . T 4—] 4—])(4—3? 
2x Ow 2) (8x)? je 
and we see that for short wave-lengths 
eae 
E, varies as 2° maar’ 2. 
For these wave-lengths Lord Rayleigh’s formula makes 
48 
Ey vary as wig 10 
and Planck’s 
4:95. 
Hi, vary as 5° ROS NE 


Thus for both long and short waves the variation of E, 
with temperature and wave-length indicated by the preceding 
theory is very much the same as that given by Planck’s 
theory ; from Aldis’s Tables the values of E, given by 
equation (2) can easily be calculated when A and @are given. 

From the equation 

2rh= a zs 
we find? putting a= 1°42 x 107 5X, 0=2940 « 1074, that 
h= ax LO 

Bat h=4,/pm. If we suppose the repulsive force due to 
an electric doublet of moment M, w=2Me, and we have 
approximately 


eG 


2Miem=10- 2*. 
or Mig? ==2:) Ones 
taking e— Fx 10 * then M= 15x 1072. 


The distance between the charges in the doublet would thus 
be 4X 107° cm. 

The existence of these doublets has a very important 
bearing on the theory of the distribution of energy in light- 
waves. There are many phenomena which can be inter- 
preted as indicating that the energy in radiation is made u 
of definite units, and that these units are indivisible, the 
energy in each unit of light of frequency being h’n/27 where 
h’ isa constant introduced by Planck, having the value 
6°55 x 10-*" erg. sec. Asan example of a phenomenon which 
suggests this division of the energy of light into definite 
units, we may quote the very interesting experiments made 


244 Sir J. J. Thomson on the 


by Ladenburg on the energy possessed by the corpuscles 
which are emitted by bodies when exposed to ultra-violet 
light. Ladenburg found that the maximum energy of these 
corpuscles was independent of the intensity of the light, 
that it varied but littie with the nature of the body from 
which they were emitted, and was proportional to n the fre- 
quency of the light, being of the order A/n/2a where h’ is 
Planck’s constant. These results admit of very straightforward 
interpretation on the unitary view of the structure of light, 
each corpuscle being regarded as taking up one unit of 
energy from the light which caused its ejection. There seems 
to me, however, to be grave objections to the assumption 
that units of light are incapable of alteration ; for example, 
why should a unit of light when passing over a corpuscle be 
obliged to communicate to it either the whole of its energy 
or none at all ? 

If we suppose that doublets exist in the atom, then 
experiments such as Ladenburg’s admit of a different inter- 
pretation from that just given. If AB is a doublet with the 
positive end at B, and P a corpuscle, then it is possible to 
have a state of steady motion when P describes a circle 
round AB as axis, the plane of the orbit being at right 
angles to AB and the centre of the orbit on the prolongation 
of AB. 

The equations of motion of a particle moving under the 
influence of the doublet are easily found. Let s=OP the 
distance of the corpuscle from O the centre of the doublet, 
@ the angle which OP makes with AB the axis of the 
doublet, and ¢ the angle which the plane POB makes with a 
fixed plane. Then m being the mass and e the charge on a 
corpuscle, M tiie moment of the doublet, we have 


dr ah cit ‘5 Me 
mT —r sin? 06? —78 =~ — “cos 6. 
m( 5 (770) —yr*’ sin @ cos 0$* )= ny sass 


d 9 * 9 
Lie (o? sin* Od) ae, | 


In the state of steady motion both r and @ are constant 
and v the velocity of the particle =r sin 0, hence we have 


, y 2 


Me sai { 


“ 


mv _ 2Me COs 6 ] 


cot @ me= 


eel 


Io 
1S 
| 


Theory of Radiation. 


Hence tan? d=2; 
HL 
tmv? = “Mme cos? @. d 
= /Mme. = 4 
= Me ais 
of 
or since h=>—— / Mme, 
v2 
/2 . 


Si pe 
4 


Thus since mv” is the kinetic energy and @¢ the frequency 
of the steady motion, we see that the steady motion of the 
corpuscle is such that the kinetic energy is proportional to 
the frequency. 

We can easily show that if the corpuscle is disturbed 
from the state of steady motion, and if ++ , tan7!/2+458, 
where p and 3 are small, are the values of p and 43 in the 


disturbed motion, then 
2 


6) 


le ae 
$/ + 2p?3 =0, 


or the frequency of the vibration about the steady motion is 
./ 2 times the frequency of the steady motion, and both are 
proportional to the kinetic energy of the corpuscle in its 
steady motion. 

By altering the distance of the corpuscle from the centre 
of the doublet always keeping tan@=2, we can make the 
frequency for steady motion any thing we please, the kinetic 
energy will always be proportional to the frequency. 

Hence, if the atoms contain doublets, it is probable that 
in a certain number of cases these doublets will have cor- 
puscles circulating round them, in some atoms the distance 
of the corpuscles from the doublet will have one value, in 
others another, and these differences in the distances will 
give rise to steady motions with different periods. Thus in 
a body made up of an enormous number of atoms, there are 
systems consisting of a doublet and a corpuscle in steady 
motion, the frequency of the motion having all values, the 
kinetic energy of this motion bears a constant ratio to the 
frequency, the frequency being independent of the kind of 
atom in which the steady motion takes place. What will be 
the behaviour of such a body when an electric wave of 
definite frequency passes through it? The electric forces 
in the wave might do work upon the doublet, twisting its 
axis so as to alter the angle it makes with the radius to the 


246 On the Theory of Radiation. 


corpuscle. From the principle of resonance the alteration 
in the angle will be far greater when the frequency of the 
steady motion of the corpuscle coincides with that of the 
incident electric wave than in any other case. A large alte- 
ration in the angle will, however, result in the corpuscle 
getting free from the doublet and going off with much the 
same icietic energy as it had in the steady motion ; this, 
as we have seen, is equal to the frequency multiplied by 
Planck’s constant. On this view then a light-wave would 
liberate corpuscles whose frequency when in a state of steady 
motion is the same as that of the light, and the kinetic 
energy of these corpuscles would be proportional to the 
frequency. Thus the energy of the corpuscles ejected by 
the light would on this view be proportional to the frequency 
of the light, whether the energy of the light-wave was 
made up of different units or not : so that we cannot regard 
Ladenburg’ s experiments as a proof of the unitary str ucture 
of light. Again, the number of atoms in which there is 
steady motion of the kind we are considering having a 
frequency nearly equal to some particular value, is probably 
a very small fraction of the whole number of molecules, so 
that the number of particles emitted would on any view as 
to the constitution of a light-wave be small compared with 
the number of molecules passed over by the light. This 
theory enables us to explain the electrical effects produced 
by light, without assuming that light is made up of unalterable 
units, each containing a definite and, on Planck’s hypothesis, 
a comparatively large amount of energy, a view which it is 
exceedingly difficult to reconcile with well-known optical 
phenomena. The existence of the doublets produces throughout 
tne body systems (the corpuscles in steady motion ) which act 
like resonators, having frequencies of all values, and pos- 
sessing an amount of energy proportional to the frequency. 

The magnetic properties of bodies sbow, I think, that the 
who'e number of these systems in steady motion for the 
whole range of frequencies cannot be large compared with 
the number of atoms. . For the corpuscle moving round the 
circle with radius 7 sin @ and velocity v is equivalent to a 
magnet whose moment is 4evr sin 0. 

We see, however, from equations (3) that vr sin @ is 
constant for all the systems and equal to 


2 Me. 
3 3 
73 YL 
so that 5 ad 1 e 
tevsin@= 33 / Mme 
“k 


Intelligence and Miscellaneous Articles. 247 


Thus, if there were but one system per atom, the sum of 
the moments of the magnets in a cubic Penimotre of gas at 
standard temperature and pressure would, taking the fa her 
of molecules in the cubic centimetre as equal to 3x 101°, be 
‘30, which is about half the value of the same quantity ‘for 
oxygen, the most magnetic gas known. 


XVIII. Notices respecting New Books. 


Les Oscillations électriques: Principes dela Télégraphie sans fil. By 
C. Tissot. Paris: Octave Doin et fils —TZhéorie des Moteurs 
Thermiques. By E.Joueuer. Paris: Octave Doin et fils. 


ee of these small books belong to the useful series published 

under the name of Encyclopédie Scientifique. 'The former con- 
sists of a very clear summary of the principles underlying the modern 
theory of electric waves and their applications. Although it is 
mainly mathematical yet the author is fully abreast of recent 
experimental work, and his mathematics is selected so as to eluci- 
date phenomena and not to display merely abstract properties 
of equations. It can be recommended as giving a thoroughly 
satisfactory presentation of the subject. 

The same attention to the practical side is paid in M. Jouguet’s 
volume on Heat Engines. Nevertheless it is only to the engineer 
who wishes to discuss scientifically the principles of his practice 
that the work will prove attractive. To such, however, the book 
will be found to contain a very thorough elementary treatment of 
the energy and heat relations of thermal transformations including 
a complete discussion of the sources of loss of efficiency. Like 
many others amongst the French School of Physicists, M. Jouguet 
is specially good on the subject of irreversible transformations. 
Concerning his numerical data, perhaps it may be pointed out that 
the retention of the old formula for the connexion of latent heat 
of steam with temperature, is liable to criticism in the light of 
more recent determinations. 


XIX. Intelligence and Miscellaneous Articles. 


To the Editors of the Philosophical Magazine. 

. The University, Glasgow, 
eae fae Ugh, 1910: 
EFERRING to Mr. W. J, Harrison’s further statement in the 

Phil. Mag. for June regarding my paper ‘‘ On the ec iees 

of Long Waves in a Rectangular Trough,” Phil. Mag. [6] vol. xvii. 

pp. 154-164, I do not admit the legitimacy of his criticism but am 

undesirous of pursuing’ the matter further. 


R. A. Housroun. 


248 Intelligence and Miscellaneous Articles. 


To the Editors of the Philosophical Magazine. 


GENTLEMEN,— 


In a paper published in the Phil. Mag. for May 1910, p. 725, 
Mr. and Mrs. Soddy and Mr. Russell describe some highly inter- 
esting experiments on the absorption of the y rays of radium by 
different materials. Since these experiments bear on some of wy 
work on y rays, I would like to make a few remarks upon them. 

According to a formula which they use, the y rays of radium 
appear to be homogeneous. I would like to point out that the 
formula used, like all the other absorption formule, can only 
approximately represent the facts. Besides the absorption of 
the rays secondary radiation of a more absorbable type than the 
primary is produced, and some primary rays probably get scat- 
tered without any change in their nature. The formula used 
does not take any account of this, and in fact it is impossible 
at present to formulate one that does. I therefore venture to 
think that the primary y rays are not homogeneous. The experi- 
ments that have been carried out on the change in penetrating 
power of the cathode rays ejected from substances exposed to 
the y rays of radium by previous screening of the rays, cannot be 
brought into harmony with the idea that they are homogeneous. 
Thus 1* have shown by means of a magnetic deflexion method 
that the proportion of slow cathode radiation to the more 
penetrating from lead exposed to y rays becomes less when 
the rays are previously passed through a metal screen. ‘This 
was also shown by Prof. Bragg and Dr. Madsent using a 
method of scattering. Mr. and Mrs. Soddy and Mr. Russell think 
that the soft cathode radiation is solely due to the soft secondary 
y rays generated in the substance by the primary rays. If that 
were so, the proportion between the cathode radiation of different 
penetrating powers would be independent of screening, but — 
according to the experiments quoted it is not. 

I have carried out some experiments on the “ scattering ” of 
y rays of different penetrating powers. These experiments are 
too lengthy to be discussed in this connexion in this letter. I 
inay mention only that the scattering of “hard” rays, obtained 
by previously passing the y rays through a lead screen, is less 
than that of the “‘ softer” unscreened rays. However, the experi- 
ments of Mr. and Mrs. Soddy and Mr. Russell should throw some 
additional light on the intricate mechanism of the absorption of 


y rays. 
Yours faithfully, 


Cambridge, May 19. R. D. Kiremay. 


* Proc. Roy. Soc. A. vol. Ixxxii. p. 128 (1909). 
+ Phil, Mag. xv. pp. 663-675, May 1908. 


Phil. Mag. Ser. 6, Vol. 20, Pl. L. 


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THE 


LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE... 


[SIXTH SERIES.] les 


AUGUST 1910. 


XX. Molecular and Electronic Potential Energy. 
By WiLu1AM SUTHERLAND *, 


. previous communications it has been shown that cohesion 
_ can be explained by the attractions between each mole- 
cule and its immediate neighbours. [or the general case of 
a homogeneous isotropic substance the most convenient 
average arrangement of the molecules to be taken as mathe- 
matically representative of that of Nature is the cubical one, 
in which case each molecule has six nearest neighbours. 
Each molecule may be regarded as an electrized sphere 
analogous to the Earth as a magnetized sphere. It has an 
electric moment, which in several of my papers is denoted 
by es, originally used to indicate the simplest electric moment, 
that of two opposite electron charges e at distance s apart. 
But es may be regarded as the symbol for the electric moment 
of the molecule, however produced. Outside of the molecule 
the electrization acts like an electric doublet of infinitely 
small axis but of finite moment es placed at the centre of the 
molecule. For the sake of simplicity and without any real 
loss of generality we may imagine the electric axis of a 
molecule parallel to one set of the edges in our assumed 
cubical arrangement. Then in order that a molecule may 
attract its six immediate neighbours it must have its electric 
axis similarly directed to those of its two axial neighbours, 
and oppositely directed to those of its four lateral neighbours. 


* Communicated by the Author. 
Phil, Mag. 8. 6. Vol. 20. No. 116. Aug. 1910. S 


250 Myr. W. Sutherland on Jolecular 


The diagram represents these relations in perspective for a 
central molecule and its six neighbours. Let. R. be: its Wise 
tance from its neighbours. Then it attracts the two axial 


ios 


neighbours with a force 6e*s?/R* and its four lateral with a 
force 3e’s?/R*. The mean attraction is 4e?s?/R*. Concerning 
the forces between the central molecule and the more remote 
ones we see that they are either repulsions or attractions 
whose average effect can be calculated. I propose to treat 
it as negligible in comparison with the attractions of the six 
immediate neighbours. The reason for doing so is this. 
The molecules of Nature are in motion, the directions of 
their axes are changing. Our cubical arrangement of the 
molecules and the assumed directions of the axes becomes a 
closer representation of the facts of Nature, the smaller the 
multiple of R to which it is extended from a central molecule. 
Even for a molecule and its six nearest neighbours at any 
instant the cubical arrangement is not a true picture. The 
real state of affairs is a succession of distorted cubical arrange- 
ments with a strictly cubical arrangement for a mean. We 
deprive our schematic representation of plasticity if we make 
the one set of dividing planes apply to a large number of 
molecules. The best way of stating the position is to say 
that near a molecule the arrangement of other molecules is 
approximately cubical at any instant, but that the accumu- 
lated effects of small departure from the strict cubical arrange- 
ment make the relations between any molecule and those 
which are not its immediate neighbours not expressible by 
means of a single cubical arrangement. Jor these reasons 
then I propose to investigate molecular potential energy on 


and Electronic Potential Energy. 25t 


the following simple principle, that a molecule attracts its 
six nearest neighbours with a force 4e’s?/R*, and that its 
effects on all other molecules may be neglected. 


1. The so-called internal molecular pressure, the Kp? of 
Laplace and the a/v” of van der Waals. 

The attraction of a molecule for one neighbour haying the 
averags value 4e’s?/R*, the attraction per unit area of matter 
lying on one side of a plane exerted on that lying on the 
other is 4¢7s?/R®. If mis the mass of a molecule and p(=1/v) 
is the density of the substance, R?=m/p, so we may write 

Kasay fe e's" p? [in Ae" s"/m30". . (1) 


2. The internal virial 4.4 >> 7f(r). 

With f(r) to denote the attraction between two molecules 
at distance x apart, this evaluation becomes easy. Let us 
make it for unit mass of the substance consisting of n mole- 
cules in volume v. For a molecule and each of its six 
immediate neighbours 7f(r) becomes 4e*s?/R3. The first > 
extends only to the six nearest neighbours, so 

rf (n)= 24e7s37/(R?. 
Then D> f(r) = 24ne*4/R*, 
422 rf (7) =6ne's?/R = 6npe’s?/m= 6n? pes? 
= 6pe*s’/m?=3Kp/2. . . (2) 


tol 


and 


3. Molecular potential energy. 


The potential energy of a molecule and one nearest neigh- 
bour for an attraction 4¢?s?/R* is 4e?s?/3R*. For the six 
nearest neighbours it is 8¢’s?/R*, and for the n molecules in 
volume v it is n/2 times this, the factor 4 being introduced 
to avoid counting the mutual energy of two molecules twice 
in the summation. So for the potential energy of unit mass 
we have 

Ane’s?/R® =4n7e’s’p=Ape’s’/m?=Kp. . . (3) 


A. Surface energy. 


First let us take the imaginary case of a substance whose 
surface is common with that of a vacuous space. A molecule 
on the surface has now only five nearest neighbours, if we 
carry our cubical scheme right up to the surface. ‘Lhe at- 
traction of one nearest neighbour would be unbalanced by an 
equal opposite force. So a process of adjustment takes place 
in the surface layer of molecules. Part of the adjustment 

re aes 


ad 


25a Mr. W. Suther!and on Molecular 


consists of evaparation into the vacuum, but for the moment 
we may neglect the effects of this. The main part of the 
adjustment < consists in a changed arrangement of the mole- 
cules in the surface layer, involving a change of density 
which is often small compared with the density because of 
the small compressibility of liquids and solids. To a first 
approximation we may say that a molecule in the surface 
layer has only 5/6 of the potential energy of a molecule in 
the body. If then we regard the potential energy of the 
whole mass as being due ‘to compression brought about by 
the attractive forces between neighbours, then we must say 
that each molecule on the surface has superposed on this 
compressional energy a tensional supply which is 1/6 of the 
potential energy of a molecule in the body. So the surface 
energy per molecule is 4e?s?/3R3 =4ne’s?/3, and the surface 
energy per unit surface often denoted by a is 4es?/3R”. 
This will apply to a liquid in contact with its vapour whose 
density is small enough to be neglected. The case in which 
the effect of the vapour becomes appreciable can be treated 
as a special one of the following. : 


5. Surface energy at the contact of two substances. 


Here the attraction between unlike molecules enters. [et 
the one substance be called 1, and have quantities assigned 
to it by the subscript 1, the other substance being 2. Then 
the average attraction between a molecule of 1 and a neigh- 
bouring molecule of 2 is 4e;5,é)5/;R»o', where ,R, has still to 
be defined. It is the average distance between the surface 
layer ot molecules of 1 and the surface layer of molecules: 
of 2. Thus the representative cubical arrangement of six 
molecules of 1 round a central molecule in the body of 1 is 
to be replaced for a central molecule in the surface layer by 
five molecules of 1 in cubical order round it, and one mole- 
cule of 2 at distance ,R, in the sixth direction. So the 
surface energy of a molecule in the surface of 1 is 4(e,?s;?/R,’ 
— €)S€089/,R,*)/3, and for a molecule in the surface of 2 it is 
A (€57s5?/Rq? — e1s1989/,R,°. Per unit surface the energies are 
A(e,7s77/Ry? — e1512,89/; Ry?) /3RY? and 4 (€,7s,7/Ro° 
— €151€285];R,°)/3R,?, so for the total surface energy or tension 
a we have 


30/4. = ey?s;"/Ry? — €181€89( 1/Ry? + 1/R,”)/, Rs? + €,°89°|Rg’. (4) 


The conditions of statical equilibrium are not provided by our 
definition of ,R, and placing of molecules in the sixth direction. 
An adjustment takes place, whose effects are neglected. 


and Electronie Potential Energy. 253 


6. Surface energy of a liquid in contact with its vapour. 


Here there is more difficulty in specifying an artificial 
arrangement which will represent the average case of Nature. 
A molecule of the surface layer of the vapour in a cubical 
scheme has five nearest neighbours amongst the vapour 
molecules and one nearest amongst the surface molecules of 
liquid, but there are many surface molecules of the liquid 
which are almost as near as the nearest, half of these re- 
pelling and half attracting. But it seems to me that it is 
right to assume that the electrical adjustments to minimum 
potential energy which cause a molecule of vapour not at a 
surface to attract its six immediate neighbours, will likewise 
cause it, when it is in a surface layer, to determine in all the 
near molecules of the liquid surface such a total electric 
moment as attracts it as if the sixth vapour molecule were 
present alone in the place of the nearest liquid molecule. 
In Nature the surface molecule of vapour is not stationary 
in the average position, but it moves right up to the liquid 
surface, establishing for itself there liquid conditions, it may 
retreat, or its place may be taken by a molecule leaving the 
liquid. Our assumption averages the values of the force 
experienced during this motion, and supposes it to be equal 
to the force between two vapour molecules whose distance 
apart is the average distance of a surface molecule of vapour 
from the liquid surface. On this supposition there is no 
reason to expect any appreciable perturbation of the vapour 
density except in the layer quite close to the liquid. It leads 
to somewhat different results from those obtained in my 
paper “ The Principle of Dynamical Similarity in Molecular 
Physics” (Boltzmann Festschrift, 1904, A. Barth, Leipzig) 
from similar considerations. By suitably changing the 
symbols in the last section with subscript 2 for vapour we 
get for the tensional surface energy of a molecule in the 
liquid 4e*s? (1/R,*—1/,R, *) /3, and for the compressional sur- 
face energy of a molecule in the vapour 4¢’s*(1/R.3— 1/,R,°)/3, 
the total per unit surface being 


a= 4e7s?{1/R— (1/R,? + 1/R,?)/,R? + 1/R,°}/3. . (5) 


It has been assumed here that es remains constant during 
the change from liquid to vapour. It is possible that two 
molecules approaching one another might alter one another’s 
electric moments by a process of induction or otherwise. 
This is a very important matter in connexion with the elec- 
trization of motecules. In section 10 we shall be investigating 
a change of equal importance within each molecule due to 


254 Mr. W. Sutherland on Molecular 


variation of its distance from its neighbours. Evidence 
bearing upon a real or apparent change of es with R will be 
discussed in the next section, the subject of the present 
section being resumed in 8. 


7. The virial of molecular attraction expressed empirically. 


In “ The Laws of Molecular Force” (Phil. Mag. [5] xxxv. 
1893, p. 211) it was shown from the extensive experiments 
of Amagat that the equation of van der Waals applies to the 
whole gaseous region of the element gases H,, O, No, and to 
CH,, down to and a little beyond the critical volume. Let 
us write that equation in its properly extended dynamical 
form for comparison with the equation of the virial of 
Clausius. It is 


3° ae Beeb J) hate 
pg be= og RI+ git lag 5 ee e e e (6) 


The term on the left is the virial of the external pressure, 
the first term on the right is the translatory kinetic energy 
of the molecuies, the second is the virial of the repulsive 
forces which act during molecular collisions, and the third 
is the virial of molecular attraction. The form of this third 
term when compared with (2) with p=1/v, shows that for 
the element gases and CH, the electric moment es does not 
vary with the distance between neighbour molecules either 
in reality or in effect. But in the same paper it was shown 
from Amagat’s experiments on CO, and from those of 
Ramsay and Young on (C,H;),0 that for typical compounds 
the equation takes empirically not the form of that of van 
der Waals, but this 


3 3 3 2k Butt - 
ppo= gRTt Blk Dake eee (7) 


‘This applies from v=o to the critical volume which is 
nearly 74/6, and it holds approximately down to v=Aé, 
Here we have two remarkable differences from the equation 
of the van der Waals type. Originally I supposed these to 
be due to a pairing of the compound molecules, but in later 
papers attributed them to molecular entanglements during 
collision. We have now again to consider them more closely. 
In the first place the virial of the repulsive forces during 
collision takes the form 2//(v+4) times, instead of b/(v~—b) 
times 3R'T/2. Now in the kinetic theory of gases v—b enters 
because under given conditions the mean free path of a mole- 
eule diminishes with increasing size of the molecule, the 


and Electronie Potential nergy. 24a 
effect of this in the dynamical calculation of the virial of 
the collisional forces is to subtract b trom v, the molecules 
being assumed to act during collision like perfectly resilient 
spheres. So the form v+/, in which & is added to v, indi- 
eates that during the collision of compound molecules, there 
is some cause which-Jengthens the mean free path, or in 
other words, reduces the frequency of collisions. A tempo- 
rary entanglement during collision is the probable cause of 
this effect. It must be remembered that though we assign a 
certain electric moment to the whole molecule, the atoms 
contribute to this electric moments of their own according 
to laws investigated in my papers on molecular attraction. 
The great distinction between the molecules of element gases 
and those of compound gases is this, that in the diatomic 
element gases the two atoms are identical. Although con- 
sisting of two atoms the molecule of an element gas, while 
not homogeneous, is more nearly homogeneous than that of 
a compound. When two compound molecules are colliding, 
the permanent electrical differences amongst the atoms may 
prevent the electric fields of the whole molecules from ad- 
justing themselves as they do when the molecules are far 
apart. It may be more an affair of the atoms in contact 
than of the whole molecule. Thus the effective moments of 
the molecules in contact may be different from what they 
are when the molecules are separate. We can account for 
the virial of the attractions taking the form 3//2(v+k) 
instead of 31/2, if we write that form (3//2v){v/(v+ &)} and 
interpret the factor v/(v+/) as expressing the change which 
takes place in the e’s? of 6pe’s?/m? in (2) with changing 
distance between neighbour molecules, that is, with changing 
v. This change in es may sometimes be partly of the nature 
of an effect of mutual induction, but it seems to me that in 
the collisional virial the change of form from v—é to v+h is 
suggestive rather of entanglement during molecular encounter, 
the atomic electric fields being thrown into a confusion which 
on the averages causes the colliding moiecules to have smaller 
total electric moments effective than when they are free. 
The equation for ethylene shows‘ the transition from the type 
for elements to that for compounds. As CH, ranges itself 
with the element gases, we infer that the electric fields of C 
and of 4H are united up into one simple field like that of 
the element gases. In the paper cited it is shown that down 
to v=k the virial of molecular attractions has the form 
](v+k) when the factor 3/2 is omitted. When v=é it is 
2k, and for values of v less than & it is J/2v. Thus in 
typical compounds / in the virial of the attractions and in the 


256 Mr. W. Sutherland on J/olecular 


attractional potential energy has for the liquid state half the 
value for the gaseous. ‘These points and many others will 
be cleared up only by a kinetic theory of liquids worked out 
as completely as the kinetic theory of gases. It is because 
of this form //(v+) involving change of / from / to //2 that 
I retain / as the symbol for a quantity. standing for the K of 
Laplace who puts p=1, and the a of van der Waals. 


8. Surface energy of a liquid and its vapour 
(continued from 6). 


Having satisfied ourselves that for the element gases and 
CH, the value of es? may be taken to be the same in the 
states of liquid and vapour, we can write for them the equation 
for surface tension « 


Ba/4 = e%s"{ 1/R,>—(1/Ry2 + 1/R,”)/, Re? +1/R.}. . (8) 


For a typical compound, if we express the various effective 
values of es in terms of that for the vapour when v=oo or 
p=0 denoted hy Ca8os then for me liquid e oe aes and for 
the vapour es?=¢,s77/(v+h) =e5,/(1 + kp), s 


3a/4 = e797 {1/2R5 — (1 vie 2 ereeoveeeenrin kp)3 
+1R5(1+kp)}. . . (9) 


Though empirically k=6v,/7 =6/7pc, where p, is the critical 
density, it simplifies matters to assume that in this connexion 
k can be replaced by 1/p., and then 

da/4 = e787 {1/2Ry>— (1/Ry? + 1/R.”)/,R.°22(1 + pp)? 
+1/R,%(1+plp.)}. (10) 
This vanishes at the critical point, as it ought. 

For the further development of this equation we can 
proceed asin the Boltzmann Festschrift, but more definitely 
and rigorously, Let us consider two typical neighbour 
molecules as regards the relative motion of approach and 
departure. Suppose one fixed while the other performs the 
relative motion. Its kinetic energy may be such as will just 
carry it to rest at infinity, or it may be more or less than 
that amount, The relative orbit may be one of infinite range 
with finite or zero velocity at infinity, or one of finite range. 
The most beautiful and familiar instances of these three 
classes of relative orbits are those described under a force 
varying inversely as the square of the distance, as in the 
case of comets under the influence of the sun. The hyper- 
bola is the orbit open at infinity on account of there being 


and Electronic Potential Energy. 204 


more kinetic energy at any point than just suffices to carry 
the comet to infinity, the parabola is the orbit of infinite 
range with zero velocity at infinity, the ellipse the orbit of 
finite range because nowhere is the kinetic energy equai to 
that acquired by falling from rest at infinity. By this con- 
sideration of orbits we can give a dynamical definition of the 
states of vapour and liquid. In a vapour the relative orbit 
of two neighbour molecules is an are of a curve of infinite 
range open at infinity, in a liquid the relative orbit is one of 
finite range. At the critical point the orbit is a transitional 
form, like the parabola, between these two, being of infinite 
range but closed at infinity. Liquefaction is the gathering 
together of neighbours of relative orbits of finite range, 
evaporation is the segregation of neighbours whose relative 
orbit is of infinite range. When a liquid is in contact with 
its vapour we have just seen from statical considerations that 
the surface layer of molecules in the liquid is in tension, as 
if at less pressure, while the surface layer of molecules of 
vapour is in compression, as if at higher pressure than 
prevails away from the surface. Let us state the facts with 


the aid of the James Thomson ABCD curve, which replaces 


gs 


Y 


the straight line AD of condensation at constant saturation 
pressure. The state of the surface layer of the liquid is 
expressed by a point between A and B, let us say at B. The 
state of the surface layer of vapour is represented by a point 
between D and C, let us say at C. The states represented 
by points between B and C are unstable. The two surface 
layers of molecules consist of subsaturated liquid and super- 
saturated vapour. he two layers might be replaced by a 
single homogeneous medium occupying the same space in a 
state represented by the point E, the instability being inter- 
preted as a continual change of state, both condensation and 
evaporation, occurring at the transition from liquid to vapour. 


, 


258 Mr. W. Sutherland on Molecular 


Dynamically we may regard this medium as consisting of 
molecules so moving that: ‘the relative orbit of two neighbours 
is a closed orbit of infinite range similar to the parabolic 
orbit of comets. Let the liquid: and vapour be at absolute 
temperature T, then the ditterence between this fictitious 


* medium of transition and the substance at the critical tem- 


perature T, is that at kinetic energy corresponding with T, 
and at density associated with ,R, two neighbour moleeuler 
in the medium could just separate to an “infinite distance 
apart and come to rest. Let ps be the density associated 
with ,;R,, then ps corresponds with that distance between 
neighbours which allows their kinetic energy proportional 
to T just to give them a relative orbit of infinite range, while 
Pc corresponds with that distance between neighbours at the 
critical point which allows their kinetic energy proportional 
to T, just to give them a relative orbit of infinite range. 
Thus the difference of the potential energy of a molecule in 
our fictitious medium of density ps, and that of a molecule in 
the critical state is equal to that of their kinetic energies. 
Let us now return to equations (8) and (10) and derive from 
them the average potential energy of a molecule amongst 
those in the surface layer of liquid and the surface layer “of 
vapour, namely 


4e%?(1/R2—2/,Re+1/R.")/6, a a 


and 
4osi{1/2Ry>—2/,R,2! (1+ p/p.)? +1/R3(1+ p/p.)}/6. (12) 


The first and the last terms taken together are the mean 
energy of a molecule in the liquid and a molecule in the 
vapour, which we may identify with the potential energy of 
a molecule in our fictitious medium of density ps. Again 
the middle term becomes the potential energy of a molecule 
at density pe, if we identify ,R,’ with m/p, in elements, and 
with m/22(1+ p/e.)2¢. in compounds. To this definition of 
,R, I have been led by the consideration that it is the simplest 
one which will give the relation discovered by Hétvés, which 
we shall obtain at once, for the last expression is now equal 
to the difference between the kinetic energy of a molecule 
at T, and at T. So, passing from molecules to gram-mole- 
cules, we have the result that the surface energy or tension 
per gram-molecule «(M/p)?* is equal to the difference between 
the translatory kinetic energy of N*® molecules (N being the 
number of molecules in a gram-molecule) at IT. and at T, 


namely 3R(T-—T)/2N1? where R is the gas constant vila 


and Electronic Potential Energy. 959 


the energy is expressed in ergs for a gram-molecule, having 
the value 
1,014,000 x 22430/273 = 83 x 10°, 


and N=2°77 x 10” x 22430. 
Thus Mine it aoer Ty. 4. (18) 


This is the relation discovered by Hoétvés (Wied. Ann. 
xxvii. 1886, p. 448) by means of the prince ple of correspond- 
ing states enunciated by van der Waals. In his experiments 
Hotvés found 2°23 to be the numerical coefficient in place of 
the 1°46 just found. for 36 normal compounds Ramsay and 
Shields found 2°121 to be the mean value of this constant of 
EKotvés. For Cl, it is 1°91, O2 1°66, and N, 1°53 (Boltzmann 
Festschrift, p. 384). The agreement between the theoretical 
coefficient and these experimental values is sufficiently close 
to justify the reasoning of this section and the assumption 
;R,2=m/p, in elements, and =m/2%9,(1 + /o-)2 in compounds, 
or in other words that at the passage from the surface layer 
of the liquid to the surface layer of the vapour the critical 
density prevails in elements, and a closely related density in 
compounds. The chief reason for the difference between 
1-46 and 2°12 is that in our reasoning, by confining our 
attention to the kinetic energy and the attractional potential 
energy, we have neglected the energies associated with the 
external pressure and with the collisional forces, that is, the 
energies corresponding with the virials 3pv/2 and 3RTv/2(v—) 
or 3RT2k/2(v+k). These approximately neutralize one 
another so long as we can use the equation pu= RT approxi- 
mately, and that is why we have been able to reason success- 
fully as if the molecules were planets and comets free from 
external force and free from collisions. At the critical point 
and near it the approximation pu=RT is too rough, whence 
the discrepancy between 1:46 and 2°12. It would lead us 
too far from the present subject to discuss the inclusion of 
these two neglected terms. ‘The chief object of the present 
section is to show how the classical statical theory of surface 
tension, developed by Laplace, Young, and Gauss, in thie 
days before the kinetic theory of matter, is connected with 
the more recent discoveries made in the light of that theory. 
Closely connected with the discovery of Hétvés is that made 
by Cailletet and Mathias (Comptes Rendus, cii. 1886, p. 1202) 
which I have discussed in the Boltzmann Festschrift. With 
temperature as abscissa and density as ordinate they ploited 
the densities of liquid and saturated vapour right up to the 
critical point, forming two branches of a curve which merged 


260 Mr. W. Sutherland on Molecular 


into one another at the critical point. When the points of 
mean density are marked they yield a straight line inclined 
to the axes. The mean density is a linear function of the 
temperature. That is the discovery made by Cailletet and 
Mathias.) 5. Young (Phil. Mag.'[5] 1 1900, pp. 293) diaz 
shown that there is a small departure from linearity. The 
relation of Cailletet and Mathias is expressed completely by 
the equation 


pit-P2—2pe2e(T.—T), . ee 


to which 8. Young adds on the right a small term in Tj —T?. 

By the principle of corresponding states ¢ is a parameter 
such that cT./p-=1, the actual values calculated by 8. Young 
ranging from 0:932 for fluorbenzene to 1:061 for ethyl 
formate. For C,H, the value rises to 1°30, and for N,O to 
1-49. For Cl, it falls to 0°7675. If we return to equation 
(10) with our interpretation of it, we can write it 


[CM Jp)? =4 (os —pe)egsgN*5]3 = 2(o, + p2—2pc)eqsgN7"]3 (15) 
~1:46 (or 2:12)(T.—1). 


Since by the principle of corresponding states we derive 
from this (»1+p2—2p-)/2p-=(T-—T)/Te which is the law of 
Cailletet and Mathias, it follows that 


é8,= 31°46 (or 2:12) TJep.N7%. 2 ee 


The law of Cailletet and Mathias is identical with that of 
Kotvos by virtue of the relations which we have adopted 
between R,, ;R2, R. on the one hand and densities on the 
other. The equations just given contain the fourth and fifth 
methods of calculating the attractional virial parameter / as 
developed in “The Laws of Molecular Force” (Phil. Mag. 
[5] xxxv. 1893, p. 211), namely, from the data of the critical 
point and from surface tension. 


9. The surface energy of mixed liquids. 


Here an interesting kinetic point is raised in connexion 
with our principle that molecules can be treated as though 
each attracted only its six immediate neighbours. Consider 
a mixture of liquids 1 and 2 containing 100 molecules of 1 
to 1 of 2. Then in a permanent uniform distribution of the 
molecules, no molecule of 2 has another molecule of 2 amongst 
its six immediate neighbours, for it is surrounded by more 
than 100 molecules of 1. In a purely statical theory with 
the assumption of permanent uniform distribution the mutual 


and Electronic Potential Energy. 261 


energy of two molecules of 2 would not enter into the ex- 
pression for the potential energy of any molecule or of an 
average molecule. 

Then again the cubical arrangement seems an unsuitable 
one to assume for a mixture of unlike molecules unless the 
volume occupied by a molecule of each is the same. Never- 
theless, by the application of kinetic principles we get over 
these two ditticulties in the following way. If p is the density 
of a mixture containing mn; molecules of 1 per unit mass, 16 
will contain mp per unit volume. It contains n, of 2 per 
unit mass and np per unit volume. Let ng, denote the 
number of molecules of 1 per unit mass in the pure liquid, 
mo. being the number for 2. Then according to the statistical 
principles used in the kinetic theory we state that the time 
for which a molecule of 1 in the mixture is one of the imme- 
diate neighbours of a molecule of 1 is the fraction np/nojp; of 
the corresponding time for the pure liquid 1. Now trom (3) 
we know that the average potential energy of a molecule of 
1 having molecules of 1 for its neighbours all the time is 
4e,’s,7/R,°, Hence the potential energy of a molecule of 1 
and the other molecules of 1 in the mixture is. 4n,pe,7s,2/Ry?no10: 
so the mutual potential energy of the n, molecules is 41)7¢,7s,p. 
In this way by making our cubical arrangement the standard 
of reference where it was geometrically possible we have 
been able to pass to the case of mixtures where it is 
impossible. 

As to the mutual potential energy of the n, and the n, 
molecules we can find it most simply by considerations of 
symmetry from the result just obtained. When n,y and n, 
are large, the number of pairs of a molecule of 1 with a 
molecule of 1 is n,?/2 nearly, while the number of possible 
pairs of a molecule of 1 with a molecule of 2 is nyng. Hence 
for the desired result there needs only to replace in 
Anje;"sy"p the ny? by 2Znyng aud e?s;" by e)5,¢282, obtaining 
8nyNe,5,e98.p. If we desire to get this from first principles we 
may return and analyse the product 4n,(e,?s,2/Ry°)(m,p/70) p1) 
in the following manner. As Ry’nq,9;=1, we have nyp/ng py 
equal to the total volume of the molecules of liquid 1 in unit 
volume of the mixture or to np/2 times the volume of a pair 
of molecules of 1 when they are neighbours. Thus the 
mutual potential energy of the n, molecules is equal to three 
times the energy of a pair of them as neighbours 4e,’s,"/R,? 
multiplied by 2/2 times the volume of a pair as neighbours, 
multiplied by n. Let ,;R, be the distance between a mole- 
cule of 1 and of 2 in their average positions as neighbours, 
then the mutual potential energy of one molecule of 1 and 


\!' 


262 Mr. W. Sutherland on Molecular 


all the molecules of 2 is 4e,s,¢89/,Rq* multiplied by ngp/2 
times 2,R,°, so for the m, molecules the factor n, is introduced 
and a factor 2 introduced because of the contrast pointed out 
above between n2/2 and n,n. Thus we arrive at the same 
result as before. lor the total potential energy of the n,+ 2, 
molecules in unit mass of mixture we have 


Ap(1 16181 + 2g€282)” = 40( pe151/1my + P2epSo/m2)”, (17) 


where p; and p,=1—p, are the masses of liquids 1 and 2 in 
unit mass of the mixture. 

The attractional virial for such a mixture is 3/2 times the 
potential energy. As to the surface energy we shall consider 
only the case where the effect of the vapour is negligible. 
Then by similar reasoning to that just used in calculating 
the potential energy of unit mass we find that liquid 1 in the 
mixture contributes the fraction (1p/np;)? of its surface 
energy per unit area as a pure liquid to the surface energy 
of the mixture, and so 


agen (nyp/ MP1) a 2(7 iyngp?| Nor?o2P py) @y2 ety? + (22,p/ N2P2) Ao 
*. a[p?=(pya?[o, + poes*[p,)’. . . « (18) 


This equation was verified (Phil. Mag. [5] xxxviii. 1894, 
p- 188; xl. 1895, p. 1) by the same experiments as proved 
the formula corresponding with 4e,s)é,s,/,R,* for the force of 
attraction between two unlike molecules. If this formula 
were to hold in a purely statical theory of surface energy it 
would imply that the distribution of the mixed sets of mole- 
cules was a purely random one. Any regular distribution 
favouring the existence of a minimum potential energy would 
be excluded. Such a result is highly improbable, and there- 
fore the formula just established may be regarded as evidence 
in favour of the active motion of the molecules in a liquid. 
This kinetic method of investigating mixed liquids has been 
neglected in the past, but it has many useful applications. 

By means of the results of this section we can explain the 
remarkable fact that so many ordinary liquids mix with so 
little contraction or expansion and so small an evolution of 
heat. Such cases as the rise of temperature on mixing water 
with sulphuric acid or with ethyl alcohol are marked excep- 
tions. For the change of potential energy on mixing a mass 
p, of liquid 1 with p,=1—~p, of liquid 2 we have 


Ap ( P1e381 | m4 -+ P2loSo] Mz)? — 40 1p 1€,"8)"] i 4D op2eo"so?| Mg”. 


In “ Further Studies on Molecular Force” (Phil. Mag. [5] 
xxxix. 1895, p. 1) it was shown that for most elements in 


and Electronic Potential Energy. 263 


their compounds, except the metals, es for the atom is nearly 
proportional to the volume of the atom, so for the molecules 
of most ordinary liquids es is proportional to the volume m/Jp. 
So the change of energy on mixing is proportional to 


4o(pr/e1 + pofo2)? rr 4p,/p1 ce 4p2|p.. 
If this is 0, then Llo=piloy + po)p2, 


which states that mixture occurs without change of volume. 
Tuus the absence of change of volume and of tuermal effect 
connected with potential energy are related. As the limiting 
volume of the molecule is proportional to its electric moment, 
and as the molecules of ordinary liquids at ordinary tempe- 
ratures occupy nearly their limiting volumes, we may conclude 
that constancy of volume on mixing and constancy of electric 
moment are connected. 


10. The relation of Mills. 


This has been discussed recently in “The Electric Origin 
of Molecular Attraction” (Phil. Mag. [6] xvii. 1909, p. 657), 
but requires to be considered further in the present connexion. 
In the simplest case the change of attractional potential 
energy for an element gas when unit mass is changed from 
liquid to saturated vapour is 4e?s*(o;—p2)/m*._ For a typical 
compound the change is 


4058, {p;|2 —pal(1 + kez) } fan’, 
or nearly 46757 {p,/2 —po](1 + po/pc) }/m?. 


The. simplest hypothesis that we could make concerning 
the internai latent heat of vaporization would be to equate it 
to these changes of potential energy. For an element gas at 
different temperatures we should have the internal latent 
heats proportional to the difference of the densities of liquid 
and saturated vapour, and for a typical compound proportional 
to p;/2 —p2[(1+ kp), or nearly to p;/2—p,/(1 +p,/p-). But the 
relation of Mills makes latent heat proportional to »,s—»p,3. 
‘he interpretation of- these results leads to an important 
principle concerning the motion of the electrons which form 
a molecule. In an investigation of the nature of dielectric 
capacity (Phil. Mag. [6] xix. 1910, p. 1) the molecule was 
treated as made of pairs of opposite electrons, each pair 
having an electric moment ec. ‘The sum of the components 
ot eo parallel to the axis of electrization of the whole atom 
or molecule forms es. For the maintenance of this state of 
affairs we must imagthe each pair of electrons in motion 


264 Mr. W. Sutherland on Molecular 


round the axis of electrization. If we imagine the axes of 
magnetization and of rotation in the Harth to coincide, it 
will furnish a large mechanical model of the pair of electrons, 
magnetization replacing electrization. The pair of electrons 
is a gyrostat electrized parallel to the axis of rotation. The 
electrons of a pair do not move round one another in a plane, 
but each may be treated on the averageas moving in a plane at 
right angles to the axis of electrization. The figure illustrates 
the conception. PN is the axis of electrization, the positive 


Fig. 8. 


electron moving in a circle round P as centre in the direction 
shown by an arrow, and the negative electron round N. 
The components of the attraction between # and p alon 
Pt and Nb keep ¢ and bp in their circular orbits, while the 
components of the attraction along NP equilibrate the forces 
acting on ¢ and p on account of the electric field of the 
whole molecule parallel to NP. The length of NP is o. 
Other rotational motions of these electric gyrostats may have 
to be considered in other connexions, but at present the 
motion postulated suffices. The independence of electric 
moment in the molecules of the element gases both of tem- 
perature and density indicates that P# and Np are small 
compared with NP, and that the rotatory energy of a pair 
of electrons js a constant like its electric moment. Probably 
the same statements apply to compounds, the change associated 
with the replacing of v in the attractional virial by v+h 
being probably apparent rather than real. Imagine an 
element gas so compressed that there is no gap between 
molecule and molecule, the whole mass being a uniform 
collection of pairs of electrons at distance r, apart, each pair 
having an electric moment eo, proportional to 7. Imagine 


and Electronic Potential Energy. 265 


the whole expanded till it fills the volume which it would 
have as saturated vapour. Let the distance between neigh- 
bour pairs be 7, and suppose the electric moment ec, now 
proportional to 72, so that the medium in its second state is 
geometrically similar to what it was in the first, then on 
account of ec, and therefore es changing in the proportion 
of r, to 7,, and therefore of R, to R, the corresponding 
molecular distances, the change of attractional potential 
energy is not proportional to py—p2 but to p,"*—p,'*. Now 
in the second state suppose that the electrons fall together in 
groups so as to form the actual molecules of the saturated 
vapour. The potential energy lost in this collapse will be con- 
verted into translatory kinetic energy of the electron pairs, for 
we have seen that their rotational energy seems to be constant. 
The relation of Mills shows that none of this kinetic energy 
appears as heat, for our imaginary operations have simply 
converted the liquid into vapour at the same temperature. 
The loss of potential energy during the imagined collapse 
has become kinetic energy required by the pairs of electrons 
to maintain dynamical equilibrium in the non-uniform state 
when they are collected in groups to form molecules. This 
kinetic energy may be regarded as internal molecular potential 
energy. When there is a change of molecular state the total 
change of potential energy is equal to the difference of the 
changes occurring when ali the electrons forming the mole- 
cules fall from one and the same imaginary uniform distri- 
butien to each of the non-uniform distributions forming a 
molecular state. The total energy required to change one 
heterogeneous distribution of pairs of electrons into another 
is equal to the differences between the changes required to 
transform the heterogeneous states into the same homogeneous 
one, it is equal to the work required to change the distance 
apart of the molecules from the one heterogeneous state to 
the other against the attraction of neighbours according to 
the inverse fourth power law, together with the supply of 
internal energy required to maintain dynamical equilibrium 
under the changed conditions of heterogeneity. The sum of 
these two quantities of energy forming the total internal 
latent heat is subject to the law discovered by Mills. This 
law could be explained by itself by supposing that each 
molecule attracts its six immediate neighbours with a force 
varying inversely as the square of the distance between them, 
and that no internal change takes place in molecules when 
their distance apart is changed. But the large mass of 
evidence gathered in my papers on molecular attraction is 
quite against this simple hypothesis, while it all supports the 


Phil. Mag. 8.6. Vol. 20. No. 116. Aug. 1910. <= 


266 Prof. A. P. Chattock on the Forces at the 


law of force expressed by 4¢;s,¢959/;Ro*.. The true significance 
of the relation of Mills seems to me to be the indication of 
broad simple dynamical law in the kinetics of electrons 
forming atoms. These ideas lead at once to the following 
speculation. 


11. The nature of chemical potential energy. 


If the view proposed in the last section is correct, namely, 
that a part of the latent heat of vaporization of a liquid is 
kinetic energy supplied to the electrons of atoms to establish 
dynamical equilibrium under changed conditions of hetero- 
geneity, it follows that the heat of chemical reactions is 
energy given out because of changed heterogeneity of the 
electrons in the reacting atoms. Is it possible that the pairs 
of electrons of two chemically combined atoms mingle like 
the molecules of two mixed liquids? Even if such mixture 
does not take place, the close approach of two different 
swarms of pairs of electrons may produce instability in the 
dynamical equilibrium of each and a fall into a new position 
of equilibrium with evolution of heat in the process. The 
internal energy of the radium atom is of the type here 
supposed to reside in all atoms as kinetic energy of the 
constitutive pairs of electrons. 


Melbourne, April 1910. 


XXI. On the Forces at the Surface of a Needle-Point dis- . 
charging in Air. By A. P. Caatrock, Professor of 
Physics in the University of Bristol™. = (> (7) (is 

(| gains strength of the field at a spherically ended electrified 

needle-point may be measured in terms of the pull of 
the lines of force upon its surface f, if the pull is due to the 
lines of force alone; a condition which is only strictly 
fulfilled when the point is not discharging. 

In 1897, while attempting to extend this method to a 
discharging point, I tried the effect of supplying the latter 
with ions of opposite sign to itself obtained from a second 
point in its neighbourhood. Some rather interesting effects 
were observed in air at atmospheric pressure ; but as at the 
time no explanation of them was forthcoming their discussion 
was postponed, and they remained unpublished. 

Recently while looking over the record of the experiments 


* Communicated by the Author. 
+ Chattock, Phil. Mag. [5] xxxii.p. 285. Young, Phil. Mag. [6] 
Xlil. p. OL2. 


Surface of a Needle-Point discharging in Air. 267 


it occurred to Mr. Tyndall that an explanation of some of 
the results had become possible in the light of modern 
theories of discharge. We therefore repeated and extended 
the old work, and an account of what has been done follows 
the present paper. This has rendered necessary a discussion 
of the question how far the pull at a discharging point is due 
to the field at its surface, and how far to purely mechanical 
forces brought about by the discharge ; and an attempt is 
here made to estimate the magnitude of these forces, and to 
show that they may be neglected in the case of our 
experiments. 


Positive Discharae from a Single Point. 

When a sharp point discharges positive electricity in air 
at atmospheric pressure it usually becomes capped with a 
luminous velvety layer, probably not more than one or two 
hundredths of a millimetre in thickness. This layer and the 
air near it is presumably the region in which ionization occurs, 
and from it therefore ions of opposite signs travel towards and 
away from the point respectively. 

In fig. 1 A represents the surface of the discharging point, 
much magnified, A and D the limits of the ionizing layer, and 
A D the axis of the point. 

Before discharge sets in the field at points 
along A D will fall off for some distance in 
nearly inverse proportion to the squares of 
the distances of these points from the centre 
of curvature of A; but on the occurrence 
of discharge some of the lines of force from 
A will end on ions between A and D, say 
at B, and others beginning on ions of oppo- 
site sign, say at O, will continue on towards 
the right, with the result that the field is 
weakened between B and C. 

At the same time changes of pressure are 
set up in the gas by the moving ions; those 
at C reducing the pressure between A and C, 
and those at B raising it upon A, so that B C is a region of 
low pressure as well as of low field intensity. 


Take first these mechanical effects of the discharge. To 
simplify the argument, suppose that there is a single layer 
of negative ions at B and another of positive ions at O ; 
and let the charge per square centimetre on B be —p and that 
on A +o. The pull per square centimetre on A due to the 
lines of force ending upon its surface is 27r¢?; and if p, be 

a 


at 


268 Prof. A. P. Chattock on the Forces at the 


the pressure excess upon A ubove the atmosphere due to the 
B ions, and p, the corresponding reduction of pressure in AO 
due to the C ions, the resultant pull per square centimetre on 


A will be 
p=210* — p+ pro, 


— p, +p thus representing the change in the pull per square 
centimetre on the point due to the current discharged 
from it if the above is a complete account of the pressure- 
producing part of the process. 

The object of what follows is to compare the magnitude of 
this change with 270”. 


Suppose first that the C ions are absent, and consider the 
effect of », by itself. 

The ions in the B layer are attracted by A, and the force of 
this attraction imparts momentum to them, some of which 


remains in the ions while the rest is transmitted to the gas 


through which they move. 

Now in the case of ordinary positive point discharge the 
B ions start very close to A, and it is safe to assume that 
both parts of the momentum end by being given up to the 
point in the form of the steady pressure p;. 1s thus equal to 
the force per square centimetre to which the B layer is sub- 
jected, viz. 4a7ap—2zp’, and we therefore have 


p=2ro?—Aropt+ 27’ 
= 27(o—p)’. 


This means that as far as their mechanical effect on the 
pull is concerned the B ions might just as well have been 
rigidly attached to A. In other words, if we attempt to 
calculate the field at the point from the observed pull upon it 
we shall obtain a value which is less than that of the field at A 
by the number of lines of force attached to the B ions, and 
which is therefore due to those lines alone which cross the 
ionizing region unbroken. 


Next consider the suction effect, p., of the C ions. 

In ordinary positive point discharge these ions also start 
very close to the surface of the metal, but they move off to 
distances which are usually large compared with the size of 
the point. Except in the region near the point, therefore, 
the momentum they give to the gas is felt as a pressure on 
any fixed plate or other bodies there may be opposite the 
point, and does not sensibly affect p. 


Surface of a Needle-Point discharging in Air. 269 


Let A in fig. 2 represent the section of a hemispherical 
point. Near its surface, discharge, when it occurs, will be 
approximately radial, and may be 
thought of as filling the cone POQ 
which has its apex at the centre of 
curyature of the point. 

If fis the field in any spherical 
layer centred at O and of radius r 
and thickness dr, the momentum 
given to this layer per square centi- 
metre per second will be 


du=fp'dr, 


assuming that ions of one sign only 
are present, and that p’ isthe volume density of the electricity 
they carry. 
Also, if V is the specific velocity of these ions, C the 
current from the point, and © the solid angle of the discharge 
cone 


Fig. 2. 


C—p 7 VOr, 
Hence 
ae = LL mes 
VOoOr 


Suppose now that the sides of the cone are impermeable to 
gas. dw will result in a difference of pressure dp between 
the two surfaces of the spherical layer such that 


dp=d 3 


and if ry is the radius of the point and the ions are all supposed 
to start from there, the pressure within the cone at the metal 
surface will be less than that at a distance r from O by the 


amount 
. Cried 
(= yo ) 


With sharp points for which 7 is a small fraction of a 
millimetre we may put r=, and obtain a value for the 
integral which is not much greater than if 7 is a millimetre 
or so, the result being an upper limit to the value of p, for 
the conditions assumed, viz. 


py=O/VO 71. 


270 Prof. A. P. Chattock on the Forces at the 


For positive discharge in air at atmospheric pressure J find 
that the field fy at the centre of a hemispherical point when 
discharge is just ceasing is given by the empirical formula 


045 
Jor)’ = constant, 


where the constant as corrected by Young (loc. cit.) is 85 if 
7) is in centimetres and fo in E.S. units *. 
Hence , 
o— 16a a. 
and M5, 24 CO00035 
Qno® = VOret ~" 


The largest current used in the experiments referred to 
was about 15 microamperes, and the largest value of 79 was 
0-062 cm. V for positive discharge in unit E.S. field is 
400 em. sec.-}, and Q was roughly 2a judging by the area 
of the glow. 

k for these data is 0°12; and, for the smallest point of radius 
0:004 em., k=0°14. 

As thearea of the point surface at which discharge occurred 
happened to coincide with that upon which etective pull 


(i. e. pull with a component parallel to the axis of the point) 


was exerted by the field, these values of & give the ratio of 
the total axial suction effect of the C ions to the total axial 
pull of the field on the assumption that both o? and p, were 
similarly distributed over the discharge area. 

o” was probably uniform (see below), but as the current 
density must have varied from zero at the edge to a maximum 
at the centre of the discharge area, p, must have varied in a 
corresponding manner. ‘lhe exact law of this variation we 
have no means of knowing, but we may obtain an idea of the 
sort of error introduced by assuming p, uniform, if we adopt 
some arbitrary law: say p, proportional to sin & (fig. 2). 

Remembering that the total suction normal to the surface 
of the point will be the same whatever the law, this particular 
law leads to an axially resolved suction equal to four-thirds that 
for uniform distribution of p,. In other words, if we take 


k=0:14 x 4/3=0-19, 


we shall correct for the want of uniformity in the distribution 
for this particular case. 


* This power of 7, and the value of the constant were obtained recently, 
and differ considerably from those given in my original paper (/oc. cit.). 
The difference is due to the tapering of the sewing-needles used in the 
earlier measurements, the effect of which upon the pull was unwarrant- 
ably neglected. The later measurements were made upon platinum 
wires with their ends rounded to hemispheres in the blowpipe. 


Surface of a Needle-Point discharging in Air. 271 


This means that if we calculate /, from P, the total resolved 
pull on the point, and assume p, uniform, 79 will be 7 per cent. 
too high; whereas if we assume the sine law it will be 9 per 
cent. too high ; always supposing of course that the values 
of k obtained above are correct. 

Actually, however, they are too high for the following two 
reasons :-— 

1. The 400 cm. sec.—! taken for V represents the specific 
velocity of fully formed ions. If the C ions do not at once 
reach their fuil size, V will be greater than 400 and k 
proportionately less. 

2. A stillstronger reason for reducing & is the fact that in 
actual discharge the surrounding gas is not kept out of the 
discharge cone as has hitheito been assumed. It is of course 
really quite free to flow in laterally, and so to prevent the 
pressure from falling in the region of the point to anything 
like the extent the above values of k suggest. Instead of 
producing a slope of pressure, the drag of the ions must be 
mainly converted into motion of the gas, and the resulting 
momentum thus transmitted to the plate rather than to the 
point. 

It seems clear, therefore, that as the error in fy due to the 
suction of the © ions is probably not much more than 
9 per cent. without either of these reductions, it will be safe 
to neglect it altogether when they are taken into account. 


Consider now the electrical effect of the discharge. 

The ionizing layer is traversed by both B and C ions. The 
B ions are densest at the side of the layer next the point, and 
the C ions at its other side. The ionizing field will therefore 
contain lines of force due to both B and C ions, none of which 
are measured by the pull; and the field calculated in terms 
of the pull is consequently too small. 

It is probable, however, that the ions are swept away so 
quickly that their lines of force form a negligibly small 
part of the field at the point. Let ¢ be the thickness of 
the ionizing layer and 7 the average density of the charge 
on the B ions, close to the metal : 


mee, esa, 
Or? Fave 
and 477t is that part of the field at the metal which is due 


to the B ions if 7 is the average value of 7 through the 
distance ¢. 


The distribution of + through t is of course unknown, but 


22 Prof. A. P. Chattock on the Forces at the 


as the effect of the B ions will be shown to be small, we may 
obtain an idea of its magnitude if we assume 7 to vary 
uniformly from its maximum value to zero in passing through 
the ionizing layer, put 477t=27t, and take for /) in the 
expression for + the value obtained from the pull, viz. 
So7 

Estimating the thickness of the glow as 0°005 cm., a 
number which is certainly too high, and assuming that this 
represents t, 2arzt is about 1 per cent. of /) for the sharpest 
point used and much less for all the others, when the current 
is 15 microamperes. 

Whether the glow and the ionizing region are exactly equal 
in extent is, however, doubtful. As already mentioned, /, for 
discharge is proportional to r°; the value of 7) therefore 
increases rapidly with the curvature of the point, and it is 
difficult to see why this should be, unless the only effect of the 
curvature upon the field (viz. the divergence of the lines of 
force) is able to influence the ionizing process. 

But for this to be, the ionizing region must reach far 
enough beyond the point to feel the divergence of the lines ; 
in other words, it seems as though ¢ should be comparable 
with the radius of the point in spite of the fact that the 
luminous region is practically confined to the surface of the 
metal. 

Yet even if ¢ is equal to 7, the field of the B ions is 
less than 3 per cent. of f, for the sharpest point and still 
less for the others ; hence when account is taken of the fact 
that the ions are newly formed and probably travel much 
faster than we have supposed, there is not likely to be any 
serious error introduced if f, calculated from the pull be 
taken as the true field at the point. 

One other effect of the discharge should be mentioned. 
The field in the discharge area is presumably constant, so 
that where discharge occurs o will also be constant, and 
Young’s correction (loc. cit.) for the distribution of o will be 
reduced. 

When the discharge area is confined to the centre of the 
point his constant must be used, and 

fyo=1:085\/ 8P/1 
but for the point under discussion glow was visible over the 
whole hemispherical end of the point from 15 down to 
1 microampere, and possibiy lower. Im all cases of positive 
discharge from this point the values of fy have been calculated 
from the formula 


io= V8P |r. 


Surface of a Needle-Point discharging in Air. 273 


It appears from the foregoing arguments that the only 
force of any importance at the surface of a positive dis- 
charging point is the pull of the field upon the metal. 
This field we should expect to be independent of the current 
from the point, at any rate for a considerable range ot 
current ; and the fact that, as the following table shows, the 
values of fy calculated from the pull are nearly constant thus 
gives considerable support to those arguments. 

In the table are given the values of “8 for various 
currents from a positive point of radius 0-018 cm. discharging 
against a flat metal plate 2°2 cm. distant. 


microamperes. * microamperes, 

7:46 0-19 7°37 13°38 
7°42 1-e4 7°38 9°39 
741 3°09 7°38 tl 
7°39 55 74L 4°20 
7°39 6-90 (ie a) 1-64 
7°39 801 743 0-79 
7°39 9°59 7-46 0 52 
7°38 10°90 793 0-0 


Negative Discharge from a Single Point. 


© for negative discharge is usually much less than for 
positive—several hundred times less in the case of the large 
point; and the glow projects into the gas to a distance 
comparable with the diameter of the point instead of being 
confined to the surface of the latter, its form varying from 
radial to trumpet-shaped. 

That O is small means that p, is large; but as the area 
affected by p, will be smail in the same proportion, these two 
effects will roughly cancel, and the only important change in 
the suction of the C ions will be due to the easier access of 
the outside gas to the cone of discharge, which implies a 
greater reduction of p, for a negative than for a positive 

oint. 
: As the glow projects into the gas the C ions start, on the 
average, further trom the point, and this also implies a 
reduction in p>». 

The B ions, so far as their mechanical effect is concerned, 
may be expected to behave much as they did for positive 
discharge, except that their momentum will not now he given 
up to the point quite so completely. 

For these reasons the pull on a negative discharging point 
is probably quite as little affected by the mechanical forces of 
the current as that on a positive point. 


274 Prof. A. P. Chattock on the Forces at the 


There is, however, an electrical effect of the negative 
discharge which requires consideration. 

The bounding surface of the discharge cone separates two 
fields—an outer one composed of lines which pass unbroken 
from the point into the gas, and an inner one composed 
partly of unbroken and partly of broken lines. 

Suppose that in fig. 2 we pass from A outwards along the 
discharge cone. The number of lines of force contained by 
the cone decreases to a minimum near the centre of the 
ionizing region. and then increases until this region is passed. 
It follows that if we draw side by side with the discharge 
cone a second similar cone in the outer field, this secend cone 
must contain, on the average, a number of lines which lies 
between the maximum and minimum numbers in the dis- 
charge cone if the outer and inner fields are to be in 
equilibrium with one another. 

Owing to the narrowing of the discharge area, and the 
increased thickness, t, of the ionizing region, the field in the 
latter, unlike that for a positive point, is chiefly composed of 
broken lines. This would result in P being far too small 
to give a correct value of 7, if the discharge area covered the 
end of the point ; but as it is, P is almost wholly due to the 
lines of the outer field. If therefore we write 


. fo=1:085 V8P/r9, 


we shall obtain a number which is less, but perhaps not much 
less than the value of the fieid at the bottom of the discharge 
cone. Except when the current is small, the measurement 
of fy for negative discharge is thus somewhat indefinite. 


Discharge between two Points. 


Suppose that to a positive point P (fig. 3) negative ions are 
sent from asecond point N in its neighbourhood, and that 
ihe average field in the ionizing layer 5 ae 
remains unaltered. The momentum effect ea 
of the N ions on P may be conveniently 
discussed under two heads. Se 

1. That of N ions which will ultimately P A 
reach the hemispherical end of P by the ee 
arrow-marked paths. \ a ® 

These will behave like C ions reversed, 
with this difference —the momentum they impart to the gas 
keeps to the cones of discharge down which they pass to A 
much more than with C ions because the sides of the point 


Surface of a Needle-Point discharging in Air. 275 


as well as its ends are receiving ions whereby the pressure of 
the gas is raised at A’A’ as wellas at A, and the lateral escape 
of gas from A is consequently hindered. 

The proportionate change (reduction) in p will thus be not 
much less than that calculated above for C ions travelling in 
gas-prooft cones, viz. 


1 ©.0°0035 
~ Veet? 


where © is 27 and C is that part of the current carried by 
N ions which reaches the hemispherical end of P. C is thus 
several times less than the whole current carried to the point 
by N ions. 

The N ions are fully formed when they reach P, so that 
400 em. sec.~! is now the correct value for V. 

In the experiments with two points, both of which were 
discharging, a current cf 15 microamperes meant of course 
a smaller current carried by N ions, and of this a fraction 
only arrived at the end of P. If we estimate this fraction at 
one-fifth and calculate an upper limit for & by assuming that 
the whole 15 microamperes were carried by N ions, the result 
is 0°016 for the largest and 0:021 for the smallest point. 

The error introduced into fp by neglecting this part 
of the momentum of the N ions is thus of the order of 
1 per cent. 

2. The remainder of the momentum received by P from 
the N ions. This is due to the wind set up by the whole 
of the N ions in passing from N to the conductors connected 
with P, instead of, asin 1,to the much smaller number which 
reach the end of P. 

It is impossible to calculate the effect of this momentum 
on P, but an upper limit was obtained by surrounding P with 
a small cage of which the wires were close enough to slield 
P electrically, but open enough to allow the wind to pass 
freely through, the wimshurst being turned at the same rate 
as in the actual experiments, and the pressure on P measured 
by tilting the apparatus. Under these circumstances P 
would be more blown against than without the cage, partly 
because the cage would attract to itself more of the N ions 
by reason of its size, and partly because the current from 
the cage would be less than from P uncovered, and so 
the wind from N would be less reduced by ions travelling 
against it. 

In no case was the observed force of the wind on P greater 
than 2 per cent. of the pull when the cage was removed, and 


276 Forces at Surface of a Needle-Point discharging in Air. 


the error in /) due to this cause must therefore have been less 
than 1 per cent. 

As,to the electrical effect of the N ions, it may be ‘suffi- 
ciently described by saying that when they enter the ionizing 
layer they behave like B ions, and before entering it they play 
the part of the fixed plate. 

And since their mechanieal effect on the pull is so small 
it follows that under the conditions of current and size of 
point considered above the conclusions already arrived at as 
to the connexion between fj and P for single discharging 
points of either sign will still hold when N ions are supplied. 


Reaction of the Electric Wind. 


It may not be out of place here to refer to the assumption 
sometimes met with in text-books and elsewhere, that the 
reaction of the electric wind is to be found atthe discharging 

oint. 
“ Reaction there must of course be—somewhere—when the 
wind is started, and its amount must be that of the momentum 
given per second to the ions; but only an extremely small 
part of it is to be found at the point itself. 

It is true that if a needle with a sharp point and its other 
end blunt be electrified until the point discharges, it will tend 
to recede from its discharging end. The electric windmill is 
a well-known instance of this. But the needle moves because 
it it pulled more strongly at the blunt than at the sharp end, 
not because it is pushed back at the latter. If the blunt 
end be electrically shielded the needle tends to come forward, 
and to about the same extent that it did before discharge 
set in; it is the shield which now exhibits reaction by its 
increased tendency to move backwards. 

But the effect on the shield is only part of the wind reaction. 
When discharge starts the distribution of electricity on all 
the surrounding conductors changes, and the electrical forces 
on them alter in such a way that the resultant of these 
alterations acts in the opposite direction to the wind, and is 
equal to its reaction. 

The wind reaction is thus to be found upon the electrified 
portions of both electrodes ; but the portion which probably 
feels it as little as any is that part of the point surface at 
which the discharge actually occurs. 


Ionizing Processes at a Point discharging in Air, 277 
Conclusions. 


When discharge occurs ata sharp point in air at atmo- 
spheric pressure, the current, dimensions, and other conditions 
being those considered in this paper, it is possible to calculate 
the strength of the field in the ionizing region at the surface 
of the point to within one or two per cent. for a positive and 
less accurately for a negative point in terms of the mechanical 
pull upon its surface; and this conclusion holds if the point 
be supplied with ions of opposite sign to itself from a second 
point in its neighbourhood. 


XXII. On the Lonizing Processes at a Point discharging in 
Air. By A. P. Caarrock, Professor of Physics, and 
A.M. Tynpatu, B.Sc., Lecturer in Physics, inthe University 
of Bristol*. ~~ | 

[Plate IV.] 


i ie explaining the phenomena of discharge at sharp points 

in gases under normal conditions, Sir J. J. Thomson 
postulates an initial ionization of a few isolated molecules in 
the gas as a preliminary to the process of discharge. 

Suppose a point to be gradually charged with positive 
electricity in the presence of these isolated ions. ‘The field 
near its surface is at first unable to do more than clear them 
away as fast as they are formed ; but as soon as it is strong 
enough to impart to the positives among them sufficient 
energy to enable these to ionize fresh molecules in their 
turn, ordinary positive discharge sets in, and a large current 
may result, accompanied by glow at the point and wind. 

In the case of a negative point the field has also to reach 
a high enough value to enable the initially formed positive 
ions to form fresh ions; but they now have the alternative 
of doing this where they bombard the surface of the metal 
instead of in the gas, and the field required is not necessarily 
so high as when gaseous molecules are to be ionized. 

For both kinds of discharge the supply of positive ions is 
pictured as kept up by ionization due to negative ions, 
these having been produced by previously formed positive 
ions and soon. Both signs of ion have therefore to be able 
to ionize as each produces the other ; and since positive ions 
require a stronger field for this than negative it is always 
' the field required by the positive ions which has to occur at 
the point. | 


* Communicated by the Authors. 


278 Prof. A. P. Chattock and Mr. A. M. Tyndall on the 


In what follows 7+ stands for the field at an electrified 
point in which positive ions are able to ionize, and f— for 
the corresponding field for negative ions ; f+ having different 
values according as the positive ions produce others in the 

gas or at the metal surface. 


Suppose now that to a charged point ions of opposite sign 
to itself are supplied in considerable quantities from some 
source in its neighbourhood. We may call such ions 
external ions. 


With a positively charged point there are three distinct 
eases that may occur :— 


(a) If the external ions find a field at the point which 
is less than f— they will simply pass to the point and 
give rise to a current from it equal to that which they 
themselves carry. 

(b) If the field lies between f— and f+, each external 
ion will produce severat more before reaching the 
point, and the current resulting may be a considerable 
multiple of that carried by the external ions. 

(c) If the field exceeds 7+ the double ionization by both 
positive and negative ions will accompany the ion- 
ization by the external ions, and a current due to 
ordinary positive point discharge w ill be added to that 
of the external ions. 


With a negatively charged point these cases reduce 
themselves to aand c, the signs + and — being interchanged : 
b does not occur because f+ is greater than f— and the 
external ions are positively charged. 


If the appearance of light at the point is to be taken as 
indicating ionization there, it will follow that for a the point 
will be dark, while for } and ¢ it will glow. 


It is to be understood in the above, that the initially 
ionized molecules are too few in number to be taken account 
of in comparison with the external ions supplied. 


These principles are illustrated in a general way by experi- 
ments on discharge between two points, made some years 
ago, which we have lately repeated and extended, and of 
which the following is an account :— 

A horizontal platinum wire P (fig. 1) with its end rounded 
to a hemisphere in the blowpipe was suspended so that it 
protruded through a hole in a vertical metal plate Q. 


Ionizing Processes at a Point discharging in Air. 279 


P and Q were both earthed, Q directly and P through a 
galvanometer. Opposite P and in the same vertical plane 
was a sharp sewing-needle N connected to a wimshurst, 


To 
Earth 


ir P 


Pe fe 
Galvan® 


f i 
Ss IES 


and so arranged that the vertical component x an the 
horizontal component y of its distance from P could be 
varied. The radius of P (0°031 cm.) was about 7 or 8 
times greater than that of N, so that N discharged more 
readily than P ; and the tendency of N to start first was 
further increased by surrounding P with a wire ring about 
11 mm. in diameter, with its plane about 7 mm. behind the 
point P. By varying x and y it was thus possible to supply 
P with varying numbers of ions from N both before and 
after P itself began to discharge on its own account. 

The end of P was viewed through a reading microscope, 
and the resultant pull, P, upon its surface was measured 
by tilting the whole apparatus so as to keep P always upon 
the cross-wire. 

It has been shown™* that under these conditions, if the 
current from the point is not greater than 15 microamperes, 
the disturbing effect of the discharge upon the pull of the 
field on the point is probably negligible compared with the 
pull itself, and that if ris the radius of the point, and fy 
the field at the centre of its surface due to the lines of force 
in the ionizing layer, 


eS sain constant, 


where the constant is 2°83 for positive, and approximately 
3°07 for negative discharge. 

In most of the earlier observations N was about 2 cm. 
long and projected from a flat plate R parallel to Q; y was 
kept at 16 cm. and 2 was varied. In the second set which 
was made with entirely new apparatus, y was varied, and 


* Chattock, Phil. Mag. pp. 272-274 of present number. 


7 i TT 


i le oe en 


oe a ees 


Soe 


ee a 


280 Prof. A. P. Chattock and Mr. A.M. Tyndall on the 


the two points were kept in line with one another, the 
radius of the point being the same as before (0°031 em.). 

In a third set points with radii lying between 0:062 and 
0-004 cm. were used. 

The results of the second set for positive and negative are 
plotted in Curves I. and II. respectively, as they were obtained 
for the widest range of conditions ; but all three sets agree 
closely in their main features. 


Positive Discharge from P. 


Curves I. (Pl. IV.) apparently exhibit all the three cases, 
a, b, c, described on p.278. Hach curve is made up of a 
steep part S, and a nearly horizontal part H, joined by a 
curve. Somewhere in this curve or below it P began to glow, 
so that along S the discharge was dark and along H luminous. 
The exact position of the beginning of the glow was not 
easy to determine, as the light nearly always grew gradually 
from small beginnings, and though the observations were 
made in the dark it was extremely difficult to tell when it 
first became visible. In this respect the yvlow differed 
markedly from that at a negative point which started 
suddenly. 

S and H correspond closely with cases a and 6 respectively. 
For besides the fact that the discharge in 8 is dark and 
in IZ luminous, the values cf fj for H rise as they should 
do when the number of external ions is reduced by increasing 
y; and fy reaches its maximum value when N is removed 
altogether (y= ), H then corresponding with case ce. 

When y=0'2 cm. fp has been reduced about 2°3 times, 
the discharge being still apparently ordinary point discharge; 
but if the points are put 0°15 cm. apart streams of small 
sparks result *. 

Provided the frequency of these sparks is not too great 
we may write 


fo= Kt, 
where ¢ is the time counted from the last spark and K is a 
constant. If P is the average pull on the point, we have 
eet awe = apparent value of / 


__ maximum value of lue of fy _ 


AEE 


* It is possible by getting the discharge to start with y greater than 
0:15, and then decreasing it to get point discharge at this distance also. 
This was done in the case of the readings discussed below. 


Ionizing Processes at a Paint discharging in Air. 281 


The maximum yalue of fy just before each spark passed is 
given by the dotted line in Curves I. for y=0Q'15 cm. 


An interesting detail was noticed in connexion with the 
position of the glaw on P in the first set of experiments. 
Here N was on one side of P and the glow always appeared 
on the side facing N, but became symmetrical when N was 
removed. This is consistent with what was said above, as 
ionization in the reduced fields of case 6 can only occur when 
the external ions approach the point. A negative paint, on 
the other hand, only glows in case ¢, and it can then 
discharge whether external ions are arriving or not; this 
agrees with the fact that the position of the negative glow 
was far less dependent upon the position of N than that of 
the positive. 


Negative Discharge fram P. 


Here, as with positive discharge, the curves consist of two 
distinct portions, 8 and H, corresponding respectively to 
dark and luminous discharge. Of these 8, as before, repre- 
sents case a, and if for a moment we neglect the curve for 
y=t0, H in every case appears to correspond with case c, 
as it should, the ordinates of the various H curves down 
to y=1'5 being roughly the same as those for no discharge 
prom N (= 00 ). 

For y=0°5 and 0°3 the curves were cut aff short by the 
passage of sparks between N and P, but there seems no 
reason to suppose that if their H portions had been obtainable 
they would have differed in position from the others for 
normal point discharge. 3 

In the case of y=0'3 the curve is shown forking, The 
reading at the top of the lower branch was taken just before 
a single spark passed, while that for the upper branch 
corresponded with a stream of sparks. The ordinate of the 
latter was therefore multiplied by a/ 3, a8 explained above, 
and this has brought it well among the rest of the H curves. 
The dotted lines correspond with the discharge of streams of 
sparks as before. | 

Between y=0°3 and y=0'15 there appears to be a funda- 
mental change in the character of the discharge. The 
ordinates of y=0'15 have been multiplied by nh 3, but this 
has not brought them anywhere near the top of y=0°3. 

It is possible that in the curve for y=1°'0 we have the 
transition stage connecting y=0°3 and y=0°15. In normal 
point discharge the glow is confined to the region near the 

Phil. Mag. 8. 6. Vol. 20. No. 116. Aug. 1910. U 


282 Prof. A. P. Chattock and Mr. A. M. Tyndall on the 


point, but in this case (y=1-0) it reached right across from 
N to P in the form of faintly luminous streamers, which 
occasionally passed into sparks for currents above about 
12 microamperes. It is true that 1:0 does not he between 
0-3 and 0°15, but the current was evidently on the verge 
of sparking ‘all along y=1:0, and a very small change in 
the conditions was probably enough to cause streamers to 
pass into sparks or sparks into streamers for y= 
and 0:3. Streamers were, in fact, once or twice oLLRe for 
y= 03. 
Values of Ionizing Fields. 


There is no indication in Curves [., as the supply of 
external ions increases, of any limit to the lowering of the 
H. portions beyond the accidental one of sparking. 

This was at first sight disappointing ; for according to the 
theory when the field at the point is below f—, it ought not 
to be possible to obtain luminous discharge, and it seemed 
unlikely that the positive and negative ions should require 
such very different fields to ionize in as corresponded with 
the highest and lowest H curves obtained. 

Now, provided there are enough external ions present to 
pr oduce a detectable amount of light, the beginnings of glow 
should occur when, and not before, the field reaches the 
value f—. 

To test this we measured fp at the moment the glow first 
became visible for a wide range of distances between P and 
N. Under these conditions it was to be expected that the 
field in which the glow was first seen would be constant and 
equal to f— so long as the supply of external ions was 
sufficient ; but that when the supply fell short the glow 
produced in this ficld, though still present, would not be 
detectable, and the field corresponding to the first wiseble 
glow would therefore be higher than f—. 

Further, the field would continue to rise as the external 
ions became fewer until it reached the value at which 
ordinary positive point discharge sets in. : 

The supply of external ions may be reduced by increasing 
either « or y. As, however, a sufficient increase of y made 
it impossible for our wimshurst to produce the highest tield 
at P, we kept y constant at 1:5 cm. and varied a. 

To limit the spreading of the ions from N this needle was 


-made to project from the flat plate R (fig. 1) and x was 


altered by moving N and R together. In this way we were 
able to reach values of fj which were practically identical 
with those obtained when N was removed from R, 


Tonizing Processes in a Point discharging in Air. 283 


The resulting values of f, are given by the line marked 
Field in Curves III. (PI. [V.), and are in good accord with 
this theory—rather surprisingly good accord considering the 
great difficulty of determining exactly when the glow started 
in the case of the observations at the lower fields. (In the high 
fields it began more suddenly and was brighter.) There is 
an obvious halting place at a field of 250 E.S. units in the 
falling of fo as # decreases. 

The fact of the field being thus constant over a certain 
range of «x values does not, however, prove that there is no 
glow below this particular field. The amount of light in 
the glow depends on the current arriving at the point and 
on the field there ; andif the current happened to be constant 
for this range of x, a constant field might merely mean that 
until this field was reached the glow was too weak to be seen, 
and not that it was absent altogether. 

To meet this objection we have plotted in Curves III. the 
current received by P at the moment the glow became 
visible. Starting with x=0 (N and P opposite one another), 
it will be seen how very far from constant this current is. It 
falls rapidly as # increases until the value of fy begins to change. 

The subsequent rise and fall of the current curve at 
higher values of « is attributable to the fact that, when z is 
comparable with y, P receives most of the current on its 
sides. For as soon as the N ions are too few to give a 
detectable glow without a higher field /o and therefore the 
current from N increases, the result being that P receives a 
larger total current than before, though its end of course 
does not. At still higher values of 2, f) becomes constant, 
and the same as for R without N ; the current now falls off 
once more as it should, and the end of P presumably receives 
no ions at all. 


Although, as already explained, if we alter y instead of x, 

we cannot trace the field curve up to the top, it is possible 
to obtain the horizontal part at f—. 
_ In Curves IV. are plotted the results of experimenting in 
this way with the same two points. The field curve becomes 
horizontal at about 240, which agrees with the 250 just 
obtained for f—, and the current curve also shows the same 
sort of behaviour as the one in Curves III. 

But the most interesting feature of Curves IV. is the way 
in which the field drops below the horizontal when N is 
brought nearer to P than about a centimetre. At this point 
the rise of the current, when y is decreased, becomes less 
marked, and when N is 0:7 em. from P the rise changes to a 


U2 


284 Prof. A. P. Chattock and Mr. A. M. Tyndall on the 


fall. This also implies a falling field ; and it thus appears 
that at a distance from their origin of less than a centimetre 
the N ions possess the power of ionizing air in fields which 
steadily decrease as this distance decreases. 

This is precisely what we should expect if the N ions take 
time to grow to their full size, and it is interesting to 
consider it in connexion with other facts bearing on the 
growth of ions. 

Franek * has shown that when discharge occurs in air 
from the sides of a fine wire in a strong field, it is extremely 
probable that the ions, whether positive or negative, do not 
reach their full size while travelling a distance of 7 mm. 
Wef have shown, by a different method of experimenting, 
that when discharge occurs from a fine point, both the 
positive and the negative ions probably travel about 3 mm. 
before they are fully formed—a result which is consistent 
with Franck’s, when it is remembered that the average 
field in the 3 mm. was probably lower than in Franck’s 
7 mm., and the ions consequently travelled slower. With 
the relatively blunt point of the present experiments the 
distance should be greater, and we now find that the distance 
of growth in the case of the negative ions seems to have 
increased to a centimetre or so. 

All these facts thus hang well together, as far as they go, 
and so afford support to the view that the negative ions do 
really take time to grow after leaving N. 

The lower limit to the size of a negative ion is the 
corpuscle. If the second drop in the field curve is really 
due to a growth of the ions, the curve ought either to become 
horizontal again when the still lower field is reached in 
which corpuscles can ionize the air, or else to cut the 
vertical axis at this field. We have made a number of 
experiments on the starting of the glow in this critical 
region, the mean of the results being given in Curve V. It 
was found impossible to bring N nearer to P than 0°14 em, 
on account of sparking, but down to this distance P could 
be made to glow in what seemed to be the normal manner. 
The numbers obtained were rather irregular, and it was 
only by making many observations that we were able to 
obtain so smooth a curve. We do not therefore wish to 
press conclusions drawn from them until we have studied 
this part of the field more carefully. As the curve stands, 
however, it certainly does show a tendency to cut the vertical 
axis at a field of about 75 E.S. units. 

* Franck, Ann. der Physik, Vierte Folge, Bd. xxi. p. 984. . 
+.Chattock & Tyndall, Phil. Mag. [6] vol. xix. p. 449 (1910). 


Tonizing Processes in a Point discharging in Air. 285 


Reference was made in the preceding paper to the fact 
that the field in which ordinary positive point discharge 
occurs depends upon the curvature of the point; and that 
in consequence of this dependence the ionizing region pro- 
bably extends a sufficient distance from the metal to feel, as 
it were, the divergence of the lines of force. 

It is interesting to find that the fields in which the glow 
first appears in the presence of N ions are similarly dependent 
upon the point. This is shown by the following Table, in 
which are given the results of experiments upon four points 
of different sizes. 


t. nze i mee Te | Alfa | Ss | alae | 
0:0619 293 84 157 19 380 ? 10 
0:0310 410 86 245 Bry 70? 6 
0:0105 662 85 325 2:0 130? 5 
00043 975 84 433 2:2 120? 8 


y is the radius of the point in centimetres; /; the field in 
which ordinary positive point discharge is on the verge of 
stopping; fe the lowest field in which glow is caused by fully 
formed N ions; and /3 the field in which corpuscles give 
rise to glow, if the views expressed above are correct. The 
values of f; are queried on account of the great uncertainty 
attending their determination. 3 

The third column illustrates the exactness of the empirical 
relation between 7, and 7, and the rough constancy of the 
fifth and seventh columns shows that 7, and /; also depend 
on 7 in a more or less similar manner. 

It must be remembered that all these fields are rapidly 
divergent, and that their values are given at the surface of 
the metal. We do not yet know the values of the weakest 
fields in which the corresponding ionizing processes can 
occur because we do not know how far the ionizing regions 
extend from the point. , 


We found that there was a certain hysteresis in the 
appearance and disappearance of the glow, especially for 
small values of y, the current having to be raised consider- 
ably before the glow would start, after which it slowly 
worked back to a minimum. At this minimum the glow 
could be made to appear and disappear by slightly increasing 


286 Prof. A. P. Chattock and Mr. A. M. Tyndall on the 


or diminishing the current strength, and it was there that 
the field was measured in each case. 

The hysteresis is perhaps connected with the fact that 
when P begins to discharge it sends + ions to N. These, 
by rendering the escape of corpuscles easier (see below), may 
inerease the average ionizing power of the negative ions 
sent to P and so diminish the field necessary for glow, and 
therefore indirectly the current. 


Ageing of the Point. 


While external ions appear to exercise little influence 
upon negative discharge from a new point the case is different 
for an old one. It is well known that when a point has 
been used a good deal it “ages” for negative discharge by 
requiring, not only a higher field to keep a given current 
flowing from it, but a field which fluctuates widely ; the 
ageing having apparently no effect upon positive discharge 
from the same point. 

In the first set of experiments thirty curves were obtained 
altogether for positive and negative discharge with and 
without N, and by the end of the nineteenth the point 
showed signs of ageing. This appears from Curves VI. 
(Pl. IV.), where the unconnected dots and circles represent 
discharges from a negative point against a plate without N.- 

Those observations made before the nineteenth curve are 
marked by the dots, and if joined up by lines give curves 
that are more or less smooth; but the circles which mark 
the later observations give curves which zigzag up and 
down in the most irregular way if treated similarly. 
Instances of this irregularity are shown in Curves VII., 
where are plotted the twenty-first and twenty-eighth curves 
taken for negative discharge without N. 

If, however, external ions are supplied to the point the 
irregularity vanishes. This is illustrated by the twenty- 
fourth curve, also plotted in Curves VII., which was taken 
with N at distances e=1°5, y=1°6 cm. 


In these earlier observations, as in the later ones of 
Curves JI., those for which external ions were supplied give 
curves which are in close agreement with one another. The 
mean position of the earlier of these curves is shown by 
the line AA in Curves VI., and may be said to follow 
approximately the line of dots, It is true that at small 
currents AA is appreciably above the dots, as it is above 
the lowest dots throughout its length, but if we allow for 


Ionizing Processes in a Point discharging in Air, 287 


the fact that the abscisse of AA are all too Jarge by the 
currents carried on the N ions and shift A A to the left the 
discrepancy becomes less marked. 


Roughly, then, it may be said that: the effect of external 
ions upon negative discharge is tu remove temporarily the 
two signs of ageing—high field and fluctuating field at 
the point; in other words, to render the old point new for 
the time being. 


Ageing has been attributed to some change in the surface 
of the point, which makes it difficult for positive ions to 
knock corpuscles out of the metal. 

Considering that a point discharging negative electricity 
produces quantities of positive ions in its neighbourhood, it 
is not easy at first sight to see why the arrival of a relatively 
small number of external positives should facilitate the 
escape of the corpuscles so much. [or the only obvious 
difference between these two sets of ions is that those produced 
at the point are newly formed, while the externals are old 
—and this ought to render the externals less able to set free 
corpuscles instead of more. 

An explanation is perhaps to be found in the following 
theory. 


The negative discharge starts in a very small spot upon 
the point surface, the glow standing out in the form of a 
luminous trumpet toa distance comparable with the diameter 
of the point, in a manner suggestive of a rush of corpuscles 
escaping through some weak spot in the surface of the 
metal. 

When the point ages a very characteristic feature of the 
discharge is observable. It is often impossible to get small 
eurrents to flow steadily. With a new point the current 
can be made to sink gradually to nothing as the wimshurst 
is slowed down ; but with an aged point it sinks gradually 
to some low finite value and then stops dead, just as if it had 
been suddenly switched off. 

In the light of this fact let us test the following hypo- 
thesis; whatever the nature of the ageing change may be, 
let its effect be such that the metal refuses to yield up 
corpuscles under bombardment by positive ions unless the 
number of these ions is considerable. 

The hypothesis is consistent with the switching off effect 
just referred to. 4! | 


288 Prof. A: P; Chattock and Mr. A: M: Tyndall on the 


It explains, what appears to be the case from Curves VI. 
and VII., that ageirig has less effect on the field at large 
currents than at small ones. 

It explains the fact that discharge will not start as a rule 
from an aged point until its electritication is far in excess of 
what is required when the point is new. For this starting 
of discharge depends on the presence of initially formed 
positive ions ; ahd if, as these are very few in number, they 
fail to obtain corpuscles from the metal, their only alter- 
native will be to ionize the gas, which of course means a 
higher field. 

Lastly it explains the effect of external ions on an aged 
point. When discharge starts in thé manner just described 
by ionizing the gas, the region on the point at which it takes 
place will be determined by geometrical conditions alone, 
and will therefore have no particular connexion with the 
place where corpuscles come out most easily. It will, in 
fact, tend to be the place where the point has been most 
used, and therefore whete they come out with greatest 
difficulty, so that even when the current is well started and 
the supply of positive ions sufficient to obtain corpuscles 
from the point we may still expeet an abnormally high field 
there: 

Now allow the initially formed ions to be reinforced by 
supplies of external ions sufficiently large to knock out the 
corpuscles freely. It will no longer be necessary to raise 
the field to that required for ionizing the gas before 
discharge will start, as the conditions for ordinary negative 
discharge will obtain. But whereas when discharge was 
started by initially formed ions alone it tended to occur at 
an aged place on the point, it now starts at the place where 
corpuscles come out easiest, siricé a large area of the point 
surface is bombarded by the external ions, and the unaged 
spots upon it are therefore sure to be discovered. The point 
should consequently behave like a new one, and this, as the 
experiments show, is precisely what happens. 

Our soméwhat arbitrary assumption, that a small supply 
of positive ions is prevented by the ageing change at a point 
from bombarding corpuscles out of it, while a large supply 
is not, thus seems to fit the facts fairly well. 


In time of course a point ought to bécome aged all over 
if persistently supplied with external ions. Wedo not know 
whether this happens or not, but it is possible that the 
beginnings of the process are to be seen in those curves of 
the first set which were taken for negative discharge with 


Ionizing Processes in a Point discharging in Air. 289 


N present. Nos. 9, 10, 13 and 14 agree with one another 
to within about 0°5 per cent., and nos. 24 and 25, the only 
others available, are practically coincident with one another, 
but 24 and 25 are about 3 per cent. higher than the four 
earlier ones. 


Relation between the Fields for Positive and 
Negative Discharge. 


The field at the outer surface of the ionizing layer at a 
positive point is the minimum in which positive ions can 
ionize. At a negative point the field at the surface of the 
metal is that required by positive ions to knock out corpuscles ; 
and if from any cause they are unable to do this there is 
still the ionizing of the gas itself open to them. It follows 
that the ionizing field at the surface of a negative point can 
never be quite as great as that at the surface of a positive 
point if the positive ions produced at each are the same. 

In Curves VI.; the line BB represents the field-current 
curve for positive discharge against a plate only, and it will 
be seen that the majority of the negative points are well 
above it. 

As explained in the preceding paper, the absolute values 

of the negative fields are not so accurately known as those of 
the positives, but it is unlikely that this will account for so 
large a discrepancy as the one in question. . 
_ We are inclined to explain it as follows :—The negative 
glow stands out a long way from the point. <A considerable 
proportion of the positive ions formed in it have consequently 
some distance to travel before reaching the point, and will 
have grown beyond their initial size when they arrive. We 
shall thus probably be dealing with older positive ions on 
the average in negative than in positive point discharge, and 
the occurrence of the stronger fields at the negative than at 
the positive point is thus reasonable. 

It is consistent with this that the field at a negative point 
becomes less, relatively to that at the same point positive, as 
the sharpness of the point increases*; for at sharp points 
the glow does not stand out so far, and as the ions thus 
have a shorter distance to go, and also move faster in 
approaching the point, they will be newer when they get 
there. 


* Chattock, Phil. Mag. [5] vol. xxxii. p. 285 (1891). 


290 Prof. Max Mason on the Flow o7 


Summary. 


1. A supply of negative ions from without to a positively 
electrified point lowers the ionizing field at its surface. 

2. Positive ions supplied to a negative point are without 
effect when the point is new. 

These two facts are shown to be consistent with accepted 
theory. 

3. A negative point may become aged with use, but 
temporarily acquires the properties of a new one when 
bombarded with positive ions. 

4. The minimum ionizing field for fully formed negative 
ions is about half, and that for corpuscles about one-seventh 
of the field in which ordinary positive point discharge takes 
place. In each case the field is measured at the surface of 
the metal. 


4 


XXIII. The Flow of Energy in an Interference Field. By 
Max Mason, PA.D., Professor of Mathematical Physics, 


University of Wisconsin™. 


‘| Pesan following investigation may answer some of the 

questions recently raised by Professor R. W. Wood f, 
regarding the lines of energy flow in a field produced by two 
similar light sources. 

The discussion will be limited, for simplicity, to the 
following case: Two points A,, A, are centres of electro- 
magnetic radiation, produced by the isochronous vibration of 
equal point charges. The direction of vibration will be 
taken at right angles to the line A,A,. Those lines of energy 
flow will be studied which lie in the plane containing the 
line A,A, and perpendicular to the direction of vibration 


§ 1. The differential equation of the lines of mean 
energy flow. 


Let r,; and r, be the distances from A, and Ay, to the 
point P (fig. 1). The electric and magnetic vectors at P 
due to the radiation from A, and from A, will be denoted 
by E,, H,; E,, H,. The vectors k, and k, are of unit 
Jengtht and have the directions from A, to P and from 
A, to P respectively; jis a unit vector in the direction of 
vibration. | 

* Communicated by the Author. 
+ Phil, Mag. 1909, xviii. p. 250. 


Energy in an Interference Feld. Z9f 
Fig. 1. 


Since j is perpendicular to k, and to ky the electric and 
magnetic vectors at P have the following values *: 


r a 
—acosn( t= =) —aeosn( t — *) 
E,= a if H,= [ k, jl, 
UG V1 
T9 Lo 
—aeosn(¢—"*) —a.cos n( ¢— =) : 
oe a a J> Bice tis Sa) Badass) toe [k, jl, 
vip ie) 


where ¢ is the velocity of light and ais a constant depending 
on the charge of the vibrators and the amplitude of the 
vibration. 

The flow of energy is determined by the Poynting vector 


Cc c 
s= re [EH|= te: H, + Hy, |. 
On writing 
acosn( t— =) = Cs acosn{ 1 = “2) =O, 
the equation 


47S fe (= a =) j, e [kj] + ° (i. | 


(5 Vy 19 


is obtained. Now 
(i, [kj] |=k,, Li, tes 


* Terms containing higher powers of 7, and 7, in the denominatois 
are disregarded. The formulas are in agreement with H. Hertz, Any. 
Phys. Chem. xxxvi. p. 1 (1888). See, e. g., Abraham, Theorie der Eleke 
trizitat, vol. ii. p. 62, or Lorentz, ‘The Theory of Electrons,’ p. 56. 


bo 


92 Prof. Max Mason on the Flow of 


and therefore 


aig for eh a |e + 


Cc 1119 To" 11" 


We are principally interested in the time mean S$ of the 
vector S, which determines a field of steady flow. By inte- 
grating over a period and dividing by the period, the following 

values for the time means of (©,’, «© ,C2, C,? are obtained : 


2 
pie k! a n IO Qr 
OO 9 Cos B ("5 —7)= 2 COS xn (1, a 11)5 


where X is the wave-length. Therefore 


aad eae! ayaa 
808 =i 1 Neel 5 Ve To) 
ao rm "7 re oe mY. 

1"2 Boayee 12 


By the aid of this expression the differential equation of 
the lines of mean energy flow may be found, i. e. the differ- 
ential equation of the curves which have pis direction of S 
at each point. Along such a curve rz may be considered as 
a function of r,. If 8; and S, denote the coefficients of k, 
and k, in the above equation, it may readily be seen (fig. 2) 


Fig. 2 


ee 
that along the curve in question 
dry: dry=8,+8, cos: 8, a8 cos 8, 


where 6 is the angle between k, and k,. The differential 
equation of the lines of mean energy flow is therefore 


2 
re’ cos O+7;? +7y7(1 + cos @) cos te r) 


dr; ro? +r) cos ?+7yr,(1+ cos 8) cos Zhe, —r;) 


‘ 


Energy in an Interference Field. 293 


§ 2. The form of the mean energy curves in che neighbourhood 
of a point. 

We shall first investigate the curves ‘“‘ microscopically,” 
examining them in a region whose dimensions are of the 
order of magnitude of the wave-length. On account of 


the great value of - all terms in the differential equation 
21 
ru 
investigation. Let 7=p1, T2=p2 be the point in whose 
neighbourhood the curves are to be studied. The differential 
equation is then 


9 
(p.” + p: cos O)drz+ pip2(1 + cos @) cos = (72—7,)(dr2—dr,) 


except cos — (7—7;) may be regarded as constant for this 


=(p;?+p.’ cos @)dry, 
and its solution is 
: eal en 
1'2(p2? + pr? cos 8) + pip2 (1 + cos 8) 97. sin = (7,—1) 


=7)(p;? + ps” cos @) + const. 


It may be assumed without loss of generality that p,—p,= md, 
where mis some integer. Then the equation of the curve 
which passes through the point 7;=p;, r2= pz is 


(p2” + pi’ cos 8) noe — (pi? +p,” cos 6) 1? =f 
2a (72—1) 


— — Pipe 
sits (1+cos @) sin ot ne 


It will be convenient to introduce rectangnlar coordinates 


Fig. 3. 


“, y as new variables in this equation, such that (fig. 3) 


T1— Py Tae ; | 
es, ASP? =y sin 0 + #003 0. 


eS a Sea os —_ 


SS 


—_— 


nr rc a 


SS OO 


mee 


at se ee 


294 Prof. Max Mason on the Flow of 


The equation then takes the form 
y sin O(p.? + p;? cos 8) + xp,?(cos” 8 — 1) 
= — MP (1+cos 6) sin 27[y sin 6~a(1—cos 8) |. 


2a 
On introducing the parameter 
a= sin @—«(1—cos 8), 
the following parametric equations of the curve are obiained: 


Lo pipx(1+cos @) sin 272 | (ps +p," cos @)a 
Qar(1—cos 0)(p.2—p?) (1—cos @)(p:2?— p;”)’ 
_ pipo(1 + cos 0) sin 27a sf pi (1+ ¢0s 0)a 
Qa sin O(p.?—py) ~ sin O(p,’—p.”) 


y 


The curves may be readily plotted from these equations. 
The figure (fig. 4) shows a set of curves of mean energy 


Fig. 4. 


flow in the neighbourhood of the point for which p./p,=4/3, 
O=7/2. (The orientation of the set with respect to the 
centres of radiation is shown in fig. 5, below.) The heavier 
straight lines give the position of the interference miniina, 
lines given by r,--7,=const., or ‘ microscopically,” by 
y cos 0+.a(cos @—1)=const. The energy thus ‘“ crinkles” 
through the field, tending to flow along the bright inter- 
ference bands, and to cut across the dark bands. 

It may be noted that the variation of the energy in passing 
from a bright interference band to a dark band decreases as 
we approach the line A,A, between A, and A,, and there are 
no interference maxima or minima on the line A,A, between 
the sources of radiation. In fact, the energy is proportional to 


E,’?+ 2(B,E,)+5,?+ H,’+ 2(H,H,)+H,; 


the vectors have the values given in § 1, and the angle 
between Hy, and Hy, is 0; so that the above quantity is (using 


Energy in an Interference Field. 295 
the abbreviations C,, C2 of § 1) 


C2? 20x Pome ant DOC. OR 
tp Se ee Et c08 04+ 
ry ry" ae) Lo MP2 2 
Ce ta CC 
ee | ber eae : FV Beoe ay 
ry i) ae) j 


The time mean of this quantity is, by § 1, proportional to 


1 1 ] 2Qar 
re peta. °°: x 2-71) (1 + cos 8). 

The truth of the above statements is seen at once from this 
expression. 


§ 3. The general course of the mean energy curves (course of 
the beam). 


It is seen from the “ microscopical” equation of the 
energy curves that the points on an energy curve for which 
?,—1r,=mr7r all lie on the straight line 


o( Ps” + p;” cos 0) =1,(p,” + po” cos A) + const., 


and the energy curve winds back and forth across this line. 
The “ general direction”? of the energy flow in the neigh- 
bourhood of the point 7;, 72 is therefore given by 


dry __r°+r,? cos 0 
dr, 1’+7r,2cos 0 


This is also the general direction of the set of curves of 
fig. 4, as a whole, 2. e. the direction of the “beam.’’ Now 
this is exactly the differential equation that would be derived 


from the expression for S in § 1 if the terms involving 
cos 5 re) were not present, 2. e. it is a curve which is 
tangent at each point to the vector 


— 1 1 
pat ae ta 


But this vector represents the velocity produced in an 
infinite liquid by two equal sources at A, and A,. The 
“general course” of a curve of mean energy flow (course of a 
“beam ”’) is therefore that of a line of flow of an incompressible 
fluid, produced by two equal sources at A, and Ag. 


oe Soe ee <== —— i 


22S 2 SE SS ee eS —— 2S xz 


CSS 


296 Dr. W. Miller on a 


In fig. 5 the curves giving the course of the beam are 
shown, “with an enlargement which indicates the “ micro- 


nS = 


scopic” form of the rays in the neighbourhood of the point 


1, /7y = 4/3, G—7/2. 


XXIV, A Constant Pressure Gas Thermometer. By 
Wiuuram Miter, .A., D.Se., Ph.D., Senior Science 
Master, Dollar Institution *. 


HE difficulties of construction of a satisfactory gas 
thermometer, either for laboratory practice or for 
refined Oe eens are well known. A correction is 
always necessary for that part of the gas which occupies the 
stem of the instrument, unless the bulb and all that part of 
the stem occupied by the gas are immersed ‘in the same bath. 
The importance of this correction increases as the temperature 
rises, and as more and more gas is expelled from the bulb 
into the stem, so that the inass of gas contained in the stem 
becomes comparable with that enclosed by the bulb. 
In colleges and schools the direct verification of Charles's 


* Communicated by the Author. 


Constant Pressure Gas Thermometer. 297 


constant pressure law is often evaded by a combination of 
the constant volume law with Boyle’s law. Apart altogether 
from the advisability of a direct verification, more parti- 
eularly as this is the form in which the law is most 
frequently applied, the latter method is quite unsuitable 
for pupils at the age when this is generally taken up in 
schools. 

The instrument described below has surmounted those 
difficulties, so far as ordinary laboratory work is concerned. 
and may also be used where even a very considerable degree 
of accuracy is required. 

The gas thermometer consists of a glass bulb ot about 
150 ¢.c. capacity connected by a siphon 8 to a graduated 
tube. During an experiment there is always in the bulb 
from 10 to 50 c¢.c. of mercury which siphons over into the 
measuring tube. In this way the expansion of the gas is 
made to take place wholly within the bulb, and therefore 
within the heater. As the gas expands more mercury is 
expelled and its volume measured. 

A capillary tube D leaves the 
top of the bulb and joins the 
siphon-tube lower down, forming 
a level indicator outside the heater 
and enabling the pressure to be 
accurately adjusted. There is never 
more than one thousandth of the 
whole volume of the gas outside 
the heater. 

By first filling the bulb com- 
pletely with mercury from the top 
of the measuring tube and packing 
the heater with ice, 100 c.c. of any 
gas may be allowed to enter the 
bulb at a temperature of 0° C. The 
volume may be measured at any 

_ temperature between 0° and 100° C. 
The calculation of the coefficient of 
expansion of the gas is greatly 
simplified by taking 100 c.c. at 0°C. 
and heating through a range of 
100 degrees. The expansion of the 
flask is to a large extent com- 
pensated by the expansion of the 

mercury, and the error particularly at 100° C. is almost 
negligible. 

Phil. Mag. 8. 6. Vol. 20. No. 116. Aug. 1910. xX 


298 ; Dr. W. Miller on a 


The following are the results for dry air obtained in an 
ordinary laboratory experiment without any Corrections being 
applied, either for the expansion of the flask or for the 
error ot the thermometer due to the part of the mercury 
thread projecting out of the heater. 


Temperature. Jixpansion. Differences. 
ON Oe mn eee UMM cH aE. 
WOM LT 3b” (Cie, 3°60 @.c 
2S eh 7°55 ¢.¢e. 3°95 c.e 
SNS Oo a ane LO Sere: 3°4576¢ 
OG) (anne LAS Si Neve: 3°83 ¢.e 
Ome eG. TSS ere: a9 Sere 
MO MOS oa Q22* iL exe: oA ee 
OS: Oka. aha 208k tee Any Vente 
BOG, Ce, DOM tigue, 3a) eee 
Ott LALO, | Or Be tnerte, 33 se 
ees SOP9) Gace 38 [ea 
36°5 


100 x 100 = (003% 


The results are shown graphically below. 


Coefficient of expansion = 


Fig, 2.—I’xpansion of Dry Air (uncorrected). 


LxPANsiON OF 1006.0. MEASURED ATO'C. 


ce) 


OC 210 


220i a 50k 0° ha SOmieaGO 70 80°) 30° tone 
TEMPERATURE. ; 


Further observations were undertaken to investigate the. 
degree of accuracy ‘attainable by the instrument. a 


Constant Pressure Gas Thermometer. 299 


‘standardized thermometer was used and a correction applied 
for that part of the mercury thread which projected outside 
the heater. Also when the stop-cock on the top of the bulb 
is left open so that the pressure inside the bulb is exactly 
equal to that outside, it is found that the mercury in the 
level indicator suffers a small capillary depression as compared 
with that in the measuring tube ; this amounted to exactly 
one of the smallest divisions on the graduated tube. By 
bringing the level indicator in front of the graduated tube 
an allowance is made for this in each reading. 

By means of a preliminary experiment the errors of 
volume due to the expansion of the flask, of the mercury, 
and to the unequal heating of the mercury outside the 
heater, etc. were determined as follows :— 

The bulb was completely filled with clean dry mercury 
at 0° C. and heated to a temperature of 100°C. The readings 
on the graduated tube at the intermediate temperatures give 
the expansion of the mercury in addition to the other errors 
mentioned above. To obtain the error in a gas-expansion 
experiment at any temperature ¢, let v be the total volume 
of gas within the bulb (obtained by adding the expansion 
found at temperature ¢ to the initial volume at 0° C.) then the 
expansion at temperature ¢t of this volume of mercury can be 
accurately determined by calculation from the known ex- 
pansion of mercury. If wesubtract this calculated expansion 
of a volume v of mercury from the reading in the preliminary 
experiment at temperature ¢, then the difference gives us 
the error of the instrument at that temperature. By a 
preliminary experiment of this kind the instrument can be 
standardized. The following are the errors thus obtained for 
the apparatus used in the succeeding investigations :— 


Final correction 


Temperature. (to be subtracted). 
BS A 0 c.c 
OE Ooo) Hera 14 ce 
7 7 at ey 18 c.c 
Be We. hs heen 23 c.c 
2.013 Ce a 21 c.c 
5 UE ORS geet 20 c.c 
HU Cher ees rey 18 c.c 
ere it Se ote 18 c.c 
Be Ges. Masai A, 16 c.c 
| CS Oe 12 c.c 
1 (UC0 ag RT RS a 07 c.e 


| 300 Dr. W. Miller on a 


tending to vanish near 100° C., and for ordinary laboratory 
work quite negligible. 


Expansion of dry air. 


Temperature. Expansion. 

OE ae alton el tee Or ere: 

IGE SS a Oe ene ae 3°8 cc 

VE LCA i Cee (oe ee 

re Noes eee 10°97 e.e 
| anal OAM sede: 14°79 e.c 
| OS oa Oe cage S30) %G.€ 
re nv ee eee 22°11 exc 
COS CS. ee eee 25°82 e.c 

SU 200. eae ae 29°24 c.c 
GOP a eae 33°36 c.e 

OS 420 > iene NG We CuCs 


Coefficient of expansion ='003669. 


Fig. 8.—Expansion of Dry Air. 


vey 50 


Evpawsionw OF 1006.c. MEASURE 
a 


3 Eities 10. 20 30°. 40° 50° 60°. .70° ~ 80 Peommemape 
TEMPERATURE. ; 


Expansion of sulphur dioxide. 


The gas was prepared by means of sulphur and strong 
sulphuric acid, and passed through a sulphuric acid drier. 


Constant Pressure Gas Thermometer. 301 


During the experiment the barometric pressure remained 
constant at 77°00 cm. 


Temperature. Expansion. 
TES RLey Me aa On" exe, 
hb SE ie 3°76 c.c 
BU... 78 ce 
Sey oe sae 13 2c 
HS 16 ee 15°06 e.c 
5S Oa 19:0 ee 
Ore o's... 22°6 cle 
My lone os. ues 26°52 c.c 
BON occ ts. sc 30°14 e.c 
Seat Ch Sie aaa 33°86 c.c 
US Ee. 37°74 c.c 


Coefficient of expansion =°003760. 


Fig. 4.—Expansion of Sulphur Dioxide. 


EXPANSION OF 100 cc. MEASURED AT O°C. 


50-0 601-7 70 | 80". 80%. fOOC 
TEMPERATURE . ‘i : 


Expansion of hydrochloric acid gas. 


The gas was prepared by means of an aqueous solution of 
hydrochloric acid and strong sulphuric acid. It was dried 


302 


by means of strong sulphuric actd. The barometric height 


A Constant Pressure Gas Thermometer. 


remained constant at 76°96 cm. 


EXPANSION OF 100CC. MEASURED AT O°c. 


Temperature. 
Oo AC, 
Oy IC; 
Bee oC. 
eure, CO. 
ees | OC 
a0, 0 
b07 1 OC 
690 > C 
1935 © 
S020 
100°4 ©. 


Coefficient of 


Fig. 5.—Expansion of Hydrochloric Acid Gas. 


Expansion. 
Se enee ane Os iere, 


Ge: agsOak oie 
cestiteie* B1796 cici 


expansion ='003741. 


Hoe io ge) 30° «40° 50 eam zo 
f TEMPERATURE. ~~ 


A graph on a small scale conveys no correct idea of the 
To test the results properly they 


regularity of the results. 


should be graphed on as large a scale as possible. 


[803° ] 
XXV. A Hydrodynamical Illustration of the Theory of tle 


Transmission of Aerial and Electrical Waves by a Grating. 


By Horace Lamps, £.R.S., and GILBERT Cook, M.Sc.* 
as theory of the scattering of aerial and electrical waves 
by 


isolated obstacles whose breadth is small compared 
with the wave-length has been discussed in a series of papers 
by Lord Rayleigh ft. A direct verification of the results is 
hardly to be looked for, but the case of a grating, which has 
been investigated by one of the present writers{, would 
appear to be more promising in this respect ; and in fact the 
transmission of Hertzian waves by a metallic grating has 
been studied experimentally, and compared with the theory 
by Schaefer and Langwitz §, and by G. H. Thomson ||, and a 
satisfactory agreement has been found. 

A confirmation of the mathematical formule may, how- 
ever, be sought in another direction. It is known { that in 
the case of a cylindrical obstacle, or system of obstacles, the 
problem is identical with that of waves ona sheet of water 
of uniform depth, as modified by cylindrical obstacles whose 
generating lines are vertical. In particular, in the longi- 
tudinal oscillations of water in a long and narrow rectangular 
tank, having one or more such obstacles near its centre, we 
have an exact analogue of aerial waves incident on a grating, 
provided the obstacles be disposed with the proper degree of 
symmetry. The effect of the obstacles in altering the period 
of the gravest mode of oscillation can in certain cases be 
calculated, and the comparison with experiment is of course 
a very simple matter. . 

The mathematical theory** may, for the purpose in hand, 
be briefly recapitulated. The origin being taken in the 
undisturbed level of the water-surface, and the axis of < being 
directed vertically upwards, we have to satisfy the equation 


Wr Oh = Oliaertea tk Pid mths eas! eet 1) 


subject to the condition that the normal derivative 66/07 
shall vanish at the rigid boundaries, and that 


a. BO (2) 


* Communicated by the Authors. 
+ Phil. Mag. [5! vol. xlii. p. 259 (1897), and vol. xliv. p. 28 (1&97) ; 
Sc. Papers, vol. iv. pp. 288, 305. 
{t H. Lamb, Proc. Lond. Math. Soc. vol. xxix. p, 523 (1898); Hydro « 
dynamics, 3rd ed., §§ 800, 301. 
§ Ann. d. Phys. vol. xxi. p. 587 (1906). 
|| Ann. d. Phys. vol. xxii. p. 365 (1907). 
4/ Rayleigh, Phil. Mag. [5] vol. i. p. 257 (1876); Sc. Papers, vol. 1, 
. 265. 
** Hydrodynamics, §§ 226, 251. 


304 Prof. H. Lamb and Mr. G. Cook on Transmission 


at the free surface (z=0), the time-factor for the simple- 
harmonic vibration being assumed to be e’. If the depth 
be h, the condition of zero vertical velocity at the bottom 
z= —h) is satisfied if we assume that ¢ involves z only 
through a factor of the form cosh k(z+h) ; and the condition 
(2) then gives 

o'=gk tanh kh. \. «=, | = 


It remains to satisfy (1), which now takes the form 


Ob | OO 1 pp 

Ve ye tb 050 on al ei 
and the condition that 0¢/dn=0 at the vertical boundaries. 
The analysis is now identical with that which applies to the 
two-dimensional form of the problem of aerial waves, or of 
electrical waves when the magnetic force is everywhere 
parallel to <. The conditions stated determine the admissible 
values of k, and the corresponding frequencies are then given 
by (3). 

Proceeding to the case of the rectangular tank, we take 
the origin at the centre of the free surface, and the axis of 
x parallel to the Jength (l). If there were no obstacies, then 
in the case of the longitudinal oscillations the second term in 
(4) would disappear, and we should have, in the anti- 
symmetrical modes, 

o=Asinka, (.°). 2). Seen 
the factors which involve z and t¢ being omitted. The con- 
dition that 0¢/d2=0 for c= +4 then gives cos $4/=0, the 


lowest. root of which is kl=a. The period is accordingly 
that of water-waves of length 2/, viz.*: 


Qa wl wh 
i oth”). . oe 


The horizontal dimensions of the obstacles being supposed 
small compared with /, the transverse component (v) of the 
velocity will be sensible only in their immediate neighbour- 
hood. We may imagine two planes «= +a’ to be “drawn, 


such that x' is moderately large compared with the dimensions 
in question, whilst still small in comparison with J. Outside 


* The verification of this formula was at one time a fay ourite lecture 
experiment of the late Sir Georye Stokes, 


of Aerial and Electrical Waves by a Grating. 305 


these, we shall have 


approximately, and therefore, for 2 >a’, 

oases Db cosike, ci fo\2 be 2) (8) 
whilst, for «< —a’, 

aan — Db COSKE A. ss S 7( 9) 


¢ being, in the gravest mode, an odd function of «. 

In the region between the planes «= +2’ the configuration 
of the lines @=const. is, on the principles explained by 
Helmholtz and Lord Rayleigh *, sensibly the same as if in 
(4) we were to put k=0. So far as this region is concerned, 
the problem is in fact the same as that of the conduction of 
electricity in a bar of metal which has the same form as the 
actual mass of water, and has accordingly one or more 
perforations occupying the place of the obstacles. The 
electrical resistance between the two planes is then equivalent 
to that of a certain length 2z’+a of an unperforated bar of 
the same section. The difference of potential between the 
two planes may be taken to be 2(kAa'+B), by (8), since ka’ 
is small ; and the current per unit sectional area is kA, 
approximately. Thus 


Bie BY = (Oe 4 ei Nak ou) oO) 


whence 
: Bares Ue ERO i se Eh) 
an 
d=A(sin ka+ fhe coska);) «| «)' G2) 
for a>’. 


The condition to be satisfied at the end «=4l gives 
cos $kl—thkasingklI=0, . . . . (138) 


which determines k. When, as in the experiments to be 
described, ka is a small quantity, this is equivalent to 


COs ll -- a) = OR wey) tae) fon ed) 


so that the intreduction of the obstacles has the effect of 
virtually increasing the length of the tank by a. 

The value of ais known in two cases. When the plane 
x=0 is occupied by a thin rigid diaphragm of breadth a, 
having a central vertical slit of breadth c, we have f 

Bias, i bavis oa 


2a 
e——— losseo— 
vis 


2a 


* Theory of Sound, § 318. 
+ Hydrodynamics, p. 512. The notation is slightly altered. 


| 
ae 
Ay 
da 
i 
i 
Hk 

| 
i 
Bis 
i 
ht 
i | 

i 


306 Prof. H. Lamb and Mr. G. Cook on Transmission 


The experiment was tried in this form, and the results were 
satisfactory so far as they went; but the motion was so 
rapidly damped that it was difficult to determine the period 
with any great accuracy. When the oscillations were started 
it was necessary to wait for some time until the turbulent 
motion of the water swirling round the sharp edges of the 
slit had subsided. 


The remaining case is covered by the formule * 


Q7xr a 
sinh —— | 
Hera ais 
PALIT ZT 
cosh=———cos —~ | 
a 
L 16 
‘a oe . e 
pad (16) 
sin ——— 
es 4a—— te At 
i: 27x 2iry | 
cosh — 2 
J 


where wy is the stream-function in the electrical (conduction 
problem, a denoting as before the breadth of the tank. The 
stream-lines y= +a correspond to the sides; for z=0 we 
have $=0, and forz =», d=x2+ 4a. The stream-line ~=0 
consists partly of an oval curve | 


. IQny 
cosh = eos 2 ee ye Sey . 
a a 2y 


which may be taken to represent the section of the obstacle, 
and partly of the portions of the axis of # which lie outside 
this oval. By assigning different values to « we obtain a 
series of possible forms. 

When the ratio «/a is small, the oval reduces to a circle 


Bap HB yy ein. ma 
approximately, provided 
a=2rUfacc. .) . 8 9 


This implies that the ratio 6/a must be smalt; but it 


appears on examination that the circular form is not seriously | 


departed from) even when a/a is a considerable fraction. 
Suppose, for example, that the transverse diameter is one- 
half the breadth, a ratio not exceeded in the actual experi- 
ments. If in (17) we put #=0, y/a=i, we find a/a=p5. 


* Hydrodynamics, p. 514. 


neti 


of Aerial and Electrical Waves by a Grating. 307 


The half-breadth of the oval in the direction of x is found by 
putting y=0; thus 


9 
cosh = 1+ 5m. gedaan «ake 


whence x/a='2537. The two diameters therefore agree 
within 14 per cent. 

The cylinders used were of circular section, and the value 
of « was calculated from the formula 


wh 
a=2btan—, Rae ea st nection lacey Gaze) 


where 6 denotes the radius; this is obtained by putting 
«=0, y=bin(17). The theoretical period was then obtained 
from (6), with (+ written for J. 

The tank used ir the experiments was 5 feet long, 8°95 in. 
wide, and about 12 in. deep. A series of observations was 
made with different depths of water, in the case of each 
cylinder, and the period compared with that obtained when 
the cylinder was removed. In order to ensure an exact com- 
parison a hook gauge was used, and the level of the water 
adjusted so as just to reach the sharp point. 

The oscillations were started by alternately raising and 
lowering one end of the tank, in an approximately simple- 
harmonic manner, by means of a lever, the period corre- 
sponding as nearly as might be to that of the free oscillations. 
In this way the production of minor surface waves was 
discouraged ; but it was found impossible, when the obstacle 
was present, to avoid altogether the simultaneous generation 
of the second normal mode of oscillation, whose period is 
(very nearly) half that of the fundamental mode which was 
the object of study. The effect of this was, however, com- 
pletely eliminated by the method used for counting the 
oscillations. This consisted in observing, by means of a 
telescope with cross-wires set up at one end of the tank, the 
reflexion of a sharply defined object on a distant building, 
the axis of the telescope being directed to a point on the 
central transverse line of the water surface. The second 
mode of oscillation referred to affects the level, but not the 
inclination, of this part of the surface ; and the modes of 
still higher frequency subsided too rapidly to affect the 
observations, which were of course only begun after a short 
interval. 

It was found possible in this way to observe as many as 
200 oscillations with an initial vertical amplitude of about 
4 in. at the end of the tank ; but it was found that greater 


308 Transmission of Waves by a Grating. 


accuracy was secured by counting only about half this 
number, owing to the occurrence of ripples due to accidental 
tremors, which interfered sensibly when the amplitude had 
become very small. The intervals of time measured in 
successive experiments under the same conditions were found 
to agree within 0:2 sec., so that the period of oscillation 
could be inferred with an error of not more than 0°002 sec. 

The results are shown in the annexed table. The first 
column gives the depth of the water beneath the point of the 
hook gauge. The second column shows the period as 
observed when there is no obstacle, and the third as calculated 
from the formula (6). The fifth and sixth columns show the 
observed and calculated periods when a cylindrical obstacle 
of the diameter indicated is introduced, the theoretical period 
being based on the formula (21), as explained. The last two 
columns show the observed and calculated increase in the 
period, due to the obstacle, in the various cases. 


| Period when Period Increase in 
Depth || no cylinder present. || Diameter ae period, 
re ee | of 
water. | | Cylinder. 
Obs. Cale. Obs. Calc. Obs. Cale. 
ins. ins. 

Teeny 2°324 2°313 4°5 2°489 2°479 165 ‘166 
a be 2°324 2313 3°45 2°413 2°401 “089 ‘088 
eg 2°324 2-318 2°21 2308 2°346 "034 033 
Sghoat | 2°324 Joe 1°59 2°338 2°329 "014 ‘016 
BAR 2-210 2221 4°5 2°364 2-380 "154 °159 
ee 2°224 pa S| 38°45 2°304 2°304. “080 083 
ee 2215 2221 Z2\) ||. 2:245 27252 030 0381 
se 2°212 DOAK P59 i) 727226 2 23T 014 ‘016 

84 2°167 pA re 4°5 2322 2:228 "155 "155 
ee | 2°167 pa Wf 3°45 || 2°249 2°254 ‘082 ‘081 
” 2°167 2 Vie Oe elie Ee 2°204 ‘O51 031 
Re 2°168 2a 1°59 2°182 2°189 014 016 
9 27112 2°108 4°5 2°258 2257 "146 "149 
ee Z°112 2°108 3°45 2°192 2°187 ‘080 ‘079 
27112 2°108 Pigye| 1 2°142 2°138 ‘030 ‘030 
ea 2101 2°108 1°59 AD 2°123 ‘014 ‘015 

94 | 2:062 2:070 4°5 221 h 2215 149 "145 
a8 2°063 2-070 38°45 2°138 2°146 ‘O75 ‘076 
5 2°062 2-070 AL 2:088 2:099 "026 "029 
15 2:062 2:070 1:59 2°075 2°084 013 014 

| i} 


A slight and variable error in level in the bed of the tank 
could not be avoided, and the depth measured at the position 
of the hook-gauge does not therefore represent quite accurately 


or 
in —_— a, eS 


x 


4 
Ny 
4 
‘ 
a 


A Galvanometer for Alternate Current Circuits. 309 


the mean depth. This circumstance accounts for the discre- 
pancies between the numbers in the second and third columns, 
but would hardly affect at all the comparison in the last two 
columns. 

The experiments could no doubt be improved upon in 
various ways, but the agreement, as they stand, between 
theory and observation seems satisfactory. It may be worth 
while to remark that a slight inclination of the bed of the 
tank, or a slight want of symmetry, or even of verticality, in 
the position of the cylindrical obstacle, would only affect the 
period by a small quantity of the second order. 


XXVI. A Galvanometer for Alternate Current Circuits. 
By W. HE. Sumpver, D.Sc., and W. C. 8. Pains, B.Sc.* 


st eae steady electromotive forces and highly sensitive 
galvanometers available for use with direct current 
tests render such tests excellent whether deflexional or 
balance methods are in question. Tests involving change of 
current, such as induction measurements, are not so satisfac- 
tory. In ballistic tests the best galvanometers are in many 
cases not sensitive enough, and though balance methods are 
available they are usually not so simple in working as those 
in which steady currents are employed. Alternate current 
tests are still less satisfactory. It is impossible to generate 
an electromotive force whose constancy is comparable with 
that of a battery or accumulator. Special difficulties arise 
owing to effects of frequency, wave-form and phase. Balance 
methods can be devised, but they are rarely of much use, 
owing to the above difficulties and to the absence of sensitive 
instruments. A distinct advance has been made during the 
last few years by the construction of improved forms of 
vibration galvanometer. But this instrument overcomes only 
some of the difficulties. It is a sensitive indicator rather 
than a measuring instrument. It must be adjusted to 
resonance for the best effects, and its sensitiveness is neces- 
sarily affected by slight changes in current frequency. 

The sensitiveness of ballistic galvanometer tests can be 
greatly increased by the use of mechanical commutators such 
as the secohmmeter of Ayrton and Perry, or subsequent 
modifications of this by Fleming and Lyle. But such 
methods only make use of a crude torm of alternate current, 
and it appears that the simplest and most effective cumulative 
method for testing effects due to changes of current, must 
in the end prove to be one involving the use of alternate 

* Communicated by the Physical Society: read June 10, 1910. 


= 


310 Dr. Sumpner and Mr. Phillips on a 


currents generated in the ordinary way. The construction of 
the indicating instrument constitutes the real obstacle. The 
reflecting instrument for alternate current circuits described 
in this paper is the result of an attempt to overcome this 
difficulty. The instrument is like a moving coil galvano- 
meter in almost every respect, except that its field is due to 
an electromagnet excited by an alternating voltage. The 
theory of non-reflecting instruments of the same type has 
already been fully explained *, but it may be convenient to 
briefly refer to it. 

If an alternating voltage V be applied to an electro- 
magnet whose winding consists of m turns, the core flux N 
will be such that 


V=rA+mN, Kon 


where 7 is the resistance of the coil and A the current 
traversing it. If the coil and electromagnet be so designed 
that for currents of the frequencies used the resistance 1s 
negligible in comparison with the impedance, we can neglect 
the term 7A. It follows that the rate of change of N will be 
at each instant a measure of V, and this will be true what- 
ever the permeability or hysteresis of the core. We thus 
have a magnet whose strength is accurately determined by 
the applied voltage whatever the physical properties of the 
core. Such an electromagnet can easily be made very 
strong. Moreover, the shorter the air-gap between the poles 
is made, the denser is the magnetic flux due to a given 
current, and the greater is the ratio of impedance to resist- 
ance for a given winding. Thus the stronger the electro- 


co) 


magnet is made by improving its magnetic circuit, the more 


accurate it becomes, provided this electromagnet is excited 
by the voltage of the circuit. 


The instrument here described has a laminated electro- 
magnet formed, of stampings shaped like figure 1. These 


* Proc. Roy. Soc. vol. 1xxx. (1908), “ Alternate Current Measurement.”” 


‘Galvanometer for Alternate Current Circuits. 311 


stampings are of two kinds—a rectangular portion with two 
straight limbs, a, 6, forming the core of the electromagnet, an | 
a specially shaped stamping, p, between the poles. The 
stamping, p, is separated from the magnet limbs on each 
side by an air-gap. Hach gap consists of two portions, a 
narrow part about 1 mm. across bounded by the straight 
edges of p, and a wider portion in which one of the vertical 
sides of the moving coil can turn round the curved edges 
of p. The moving coil, of 50 turns, is similar in shape, 
suspension, and mode of control to that of an ordinary 
permanent magnet instrument. It has a central spindle 
indicated at s and working in a recess suitably stamped in p. 
The pile of stampings is about 4 cm. deep and 9 cm. long, 
the limbs being 1 cm. wide. Hach of the limbs, a, 3, 1s 
wound with a coil of 2000 turns of fine wire, and also with a 
coi! of 100 turns of thicker wire. The two fine wire coils 
are put in series and connected to three terminals; the two 
thicker coils are also put in series and joined up to two 
additional terminals. ‘Thus the instrument can be excited by © 
a winding consisting of either 200, 2000, or 4000 turns, 
according to the voltage used. The iron is not too strongly 
magnetized if the winding used contains 20 turns per volt on 
50 eycle circuits. Thus 200 volts may be applied to the 4000 
turn coil, or 10 volts to the 200 turn coil. But the instru- 
ment is so sensitive that such excitation will only be needed 
for exceptional tests. The moving coil may be used with a 
condenser or other apparatus, either on some special circuit, 
or in conjunction with one or other of the field coil windings. 
The instrument has been constructed by Robt. W. Paul, to 
whom several of the working details are due. The following 
are some of the uses :— 


Use as a Voltmeter. 


If a voltage V be applied to a field coil of m turns and if 
another voltage V, be applied through a condenser of K 
microfarads to the moving coil, it can be shown that the 


torque acting on the moying coil is a measure of 


a 


m 


K VV, Nyy ek aah M8 


or of the mean product of the two voltages. If the con- 
denser voltage is obtained from one of the field coils of n 
turns the torque is measured by 


is : 
a a fe WO) 


312 Dr. Sumpner and Mr. Phillips on a 


There is really another factor the value of which would not 
be quite constant if the induction density in the gap varied 
with the position of the moving coil. But in the present 
instrament this factor is essentially constant owing partly to 
the shape of gap adopted, and partly to the fact that fora 
reflecting instrument the movement of the coil is very slight. 
Numerous tests have shown that for any given choice of coils 
and condenser, the scale deflexion is strictly proportiona! to 
the square of the applied voltage quite up to the limits of the 
scale used (300 mm. each side of zero for a scale distance of 
1 metre). 

The numbers denoted by m and n may each be chosen 
either 200, 2000, or 4000, while the capacity K may be 
given widely different values. It is thus clear that the 
instrument can be used as a voltmeter for a large number of 
ranges. It will be sufficient to indicate two of these. From 
a number of tests made under various conditions, the value of 
expression (3) when V is measured in volts and K in micro- 
farads is found to be 1°6x10-* for a scale deflexion of 
200 millimetres. It follows that this deflexion can be 
obtained 


for 200 volts if m= 4000, n= 200, and K=3:2 x 10-4m.f., 
or for 20 millivolts if m=200, n=4000, and K=4°0 m/f. 


The deflexion is independent of frequency and wave-form 
if the field winding to which the voltage is applied has a 
resistance negligible in comparison with its impedance. This 
will always be the case if the frequencies used are high. But 
if the frequency is low and the mass of copper used in the 
magnetizing coil is small, the resistance of this coil will 
become comparable with its impedance. The deflexion will 
then be dependent on frequency, though for a given frequency 
it will still be a measure of the product K V*. Thus if a coil 
of two turns be wound round the core of the magnet and be 
used as the exciting winding (m=2), and if the moving coil 
be used with the same condenser and field winding as in the 
second case above (n=4000, K=4), a deflexion of 200 mm. 
will correspond with a reactive voltage in the two-turn coil | 
of only 0:2 of a millivolt. But it will be necessary to apply 
a much greater voltage than this to cause the magnetizing 
current to flow through the resistance of the winding. The 
instrument will still act as a voltmeter for constant frequency 
circuits, but its indications will be sensitive to change of 
frequency. 

With the instrument as actually wound, the effect of 


Galvanometer for Alternate Current Circuits. 313 


frequency can be represented by the measured values of the 
quantity (3) for a deflexion of 200 mm. If this quantity 
when multiplied by 10,000 be called Q, then for the arrange- 
ment m=4000, »=200, the value of Q is 1°62 for 50 cycle 
circuits, 1°61 for 100 cycle circuits, and 1°73 for 25 cycle 
circuits. That is, the deflexion for a given value of V® is 
essentially the same for all frequencies above 50 cycles per 
second, but is 64 per cent. less if the frequency is dropped 
to 25 cycles per second. For the arrangement m=200, 
n= 4000, a change of frequency produces greater effect. The 
deflexion for a given value of V? is 5:2 per cent. greater for 
100 cycles, and 186 per cent. less for 25 cycles, than it is for 
50 cycles. When the main coil is used for both voltage and 
condenser (m=n=4000) there is no appreciable change of 
constant for frequencies between these limits. 

The effect of frequency on the value of @ is mainly due to 
the phase error represented approximately by the ratio of 
resistance to impedance of the coil to which the voltage is 
applied. This ratio on 50 cycle circuits is 2°6 per cent. for 
the 4000 turn coil, and 10°6 per cent. for the 200 turn coil. 
But Q is also affected by slight amounts of magnetic leakage 
(between the windings) dependent on the arrangement of 
coils used. Other properties of the magnet are deducible 
from the data that on 50 cycle circuits the power factor of 
the magnetizing coil is 0:14 ; the ampere turns needed for the 
magnet are 6u; and the flux density in the iron is 80u ; 
where wu is the number representing the voltage applied per 
1000 turns, or the millivolts per turn. For special uses of 
the instrument the phase error of the magnet can be reduced 
by applying a suitable condenser direct to one of the field 
windings. Thus the power factor of the 200 turn coil can 
be raised to unity on 50 cycle circuits by applying a condenser 
of 1:1 m.f. to the 4000 turn coil, the ratio of resistance to 
impedance is reduced from 10°6 per cent. to 1:5 per cent., 
and the phase error is reduced to zero. : 


Use with Null Methods. 


Figs. 2--7 illustrate the ordinary bridge methods for com- 
paring inductances and capacities. In these methods a 
ballistic galvanometer is used as an indicator, and to test the 
balance the current A through the arms of the bridge is made 
or broken by a key. The equation representing the condition 
for inductive balance is indicated in each case beside the 
figure. ‘The zero deflexion condition for steady currents 


Phil. May. 8. 6. Vol. 20. No. 116. Aug. 1910. ¥ 


314 Dr. Sumpner and Mr. Philips on a 


holds necessarily in the three cases of figs. 2,4 and 6; but a 
troublesome special adjustment is needed in the case of fig 3; 
while in the cases of figs. 5 and 7 the adjustment for steady 


R, R, 


Livi eis, 2 
Ryerss, 


Fig. 4. 


R, R, 


M, Me M, 
Ri+r, R,+7r, 


currents is impossible. In these figures capacities, self- 
inductances, and mutual inductances are respectively denoted 
by the letters K, L, M; resistances are indicated by the 
letters R, 7, and 8; while the indicator is denoted by m.c.. 


Galvanometer for Alternate Current Circuits. d15 


All these methods may be used with the present instru- 
ment for steady inductive balances on alternate current 
circuits, and the same formule apply to the zero deflexion 


condition, provided (i.) the alternate voltage V applied to the 

field-coil of the instrument also causes the current A through 

the bridge conductors, (ii.) the alternate current A is made 

essentially cophasal with V by the use of suitable non- 
Y 2 


316 Dr. Sumpner and Mr. Phillips on a 
9 lines in the 


inductive resistances as indicated by the zigzag 

figures, (iii.) the moving coil m.c. of the instrument is placed 
dir ectly across the bridge (using a reversing key when 
desirable). 

It results from the special properties of the instrument 
that the flux in the gap of the electromagnet is in quadrature 
with the applied voltage (and thus in quadrature with A). 
The inductances or capacities produce voltages or currents 
also in quadrature with A, and thus in phase with the flux, 
so that their phase is such that they produce the maximum 
torque on the moying system. 

‘hese methods have all been thoroughly tested on alternate 
current circuits with the present instrument, and with most 

satisfactory results. The balance can be adjusted with ease 
to one part in 10,000, when the voltages set up on the coils 
or condensers are merely of the order of one volt, and thus 
suitable for use with the resistance boxes or dinarily found in 
laboratories. 

Certain special points call for notice. When a balance of 
great precision is needed, the minute electromotive force e, 
induced in the moving coil by the alternating field of the 
magnet, tends to cause a small deflexion disturbing the 
balance. When the moving coil circuit is essentially non- 
inductive, as for the cases of figs. 3,4 and 5, the current due 
to e will be in phase with e, and in quadrature with the flux, 
and in such cases the corresponding deflexion will in general 
be negligible. For the inductive circuits represented in 
figs. 2, 6 and 7, this will not be the case, and a small de- 
flexion due to e will occur. Butin all cases any effect due to e 
can be accurately eliminated by using a false zero method, 
that is, by adjusting the balance till the reading on the scale 
is unaltered by switching the bridge current “A on or off, 
The induced voltage e is “due to the voltage applied to the 
field coil, and is unaffected by changes in A. In most 
cases it will be found sufliciently accurate to take the mean 
of the two conditions of balance obtained by using a 
reversing key with the moving coil. ‘The false zero method 
is simpler and is mathematically accurate, though in prac- 
tice, as with all false zero methods, there is a liability to 
a small error due to the variations of the false zero deflexion 
in sympathy with fluctuations in the main current or 
voltage. 

The formula given for balance expresses the necessary and 
sufficient condition that the two electromotive forces set up in 
the coils, or on the condensers, of the bridge, send, through 
the moving coil, currents which are equal in magnitude and 


ey, 


Galvanometer for Alternate Current Circuits. 317 


exactly opposite in phase*. But if this condition is not quite 
fulfilled, the unbalanced current will not necessarily be in the 
best phase to influence the deflexion unless certain limitations 
are borne in mind. The resistances R must not be made too 
small, and the resistances S must not be made too large. 
Otherwise the sensitiveness of the instrument to indicate want 
of balance is adversely affected, although the condition of 
balance remains as stated, except for minute correction terms, 
due to secondary effects of self-induction, &¢., which have 
been neglected. 

We have found on investigation that the only cases which 
need be considered are those in which condensers are used 
(figs. 2, 6and7). The value of KSp (where p is 27 times 
the frequency) represents the tangent of the angle by which 
the phase of the moving coil current differs from that of the 
magnet field. It may e easily become comparable with unity, 
as will be apparent from the fact that on 50 cycle circuits 
with K equal to 1 microfarad, and 8 equal to 1000 ohms, the 
value of KSp is 0°314. But it will be found easy to adjust 
the conditions of the bridge in all the cases considered so as 
to render these tests quite satisfactory in practice. 

One or two examples of these bridge methods may be given 
to illustrate the conditions of working. 

The method of fig. 6 was used to test the values of M for 
a primary coil of 500 turns in conjunction with two secondary 
coils. The three coils were wound on a wooden bobbin and 
the primary wire was suitable for acurrent of 2amperes. A 
current of 1-1 ampere was passed through the bridge, and 
28 volts were applied to the 4000 turn coil, the frequency 
being 50 cycles per second. A standard resistance of 
0:9995 ohms was used for S, and a standard mica condenser 
of 1:0155 m.f. was used for K. An ordinary resistance-box 
was used for R. Using the first secondary coil the vaiue of 
R+~+r was adjusted to 4225-Lohms. The cor responding value 
of M works out to be 4°2885 millihenries. A similar test 
with the other secondary yielded 3°8499 m.h., and one with 
the two secondaries in series yielded 8°1398 m.h. The sum 
of the values of M for the two secondaries is 8-1384 m.h. 
The small inconsistency is easily attributable to errors in the 
resistance-box, to small capacity effects in these resistances, 


* This is strictly true for the cases of figs. 2,4, and 6. In the cases 
of figs. 3,5, and 7 an additional current through the moy ing coil is caused 
by the resistance of the inductance coils. This current is in quadrature 
with the field, and does not cause any deflexion. There is thus no need 
for a troublesome double adjustment as in corresponding tests in which 
other instruments are used. 


EE ee 


iam 


318 Dr. Sumpner and Mr. Phillips on a 


or to similar causes of no present interest, the point being 
that it was possible to adjust R to one part in 40,000, under 
conditions of test which could easily have been rendered more 
sensitive. The current used with the bridge could have been 
quadrupled, and the strength of the field could have been in- 
creased ten times, without injuring the apparatus, and with- 
out altering the quantities under test. 

The method of fig. 2 was used to compare the capacity of 
a paraffin paper condenser (about 0°9 m.f.) witb the standard 
mica condenser just referred to. The 4000 turn field-coil 
was subjected to 20 volts on a 50 cycle cireuit. From this 
voltage was obtained, by means of a small transformer, a 
cophase voltage of 2°5 volts suitable for the bridge conductors. 
The resistances S were kept below 2000, and various tests 
were made. It was always possible to adjust the balance to 
1 part in 10,000, but the inconsistencies in the various tests 
amounted to 2 parts in 1000 and are attributable to pheno- 
mena (such as a partial conduction in the paper condenser) 
affecting the exact formula for balance. 

An adjustable air-condenser, formed of a fixed and moving 
set of plates like a multicellular voltmeter, was tested for 
capacity against the standard mica condenser above referred 
to. The maximum capacity was measured as 0:0023 m-f., or 
only 0-2 per cent. of that of the standard. .It was always 
possible to measure the capacity in any position far more 
accurately than the condenser could be adjusted to this posi- 
tion. The method of fig. 2 was used, but higher voltages 
were applied to the bridge than in the previous test. The 
4000 turn field-coil was subjected to 30 volts, and this voltage 
was also used for the condenser bridge, a resistance of 9000 
ohms, made for the pressure circuit of a wattmeter and 
suitable for high voltages, being used in association with the 
air-condenser. The standard mica condenser was used as K,. 
The associated resistance 8, was taken from an ordinary 
resistance-box. It was set at various values up to 22 ohms 
asamaximum. The 9000 ohm resistance was used for Sg, 
and the air-condenser K, was adjusted for each value of 8, 
till balance was obtained. The maximum voltage to which 
S, was subjected in these tests was less than O0°1 volt. The 
false zero method was employed for balance, the moving coil 
voltage causing a deflexion of about 10 centimetres. 

It is to be noted that in all these bridge tests the phase 
error of the electromagnet due to the resistance of the field 
winding does not lead to any error, but merely causes a 
negligible change of sensitiveness due to a shift of phase of 
the moving coil current as compared with that of the field. 


Galvanometer for Alternate Current Circuits. 319 


Oiher Uses of the Instrument. 


The voltmeter tests previously described show that the 
instrument can be used to measure very small capacities, 
especially in cases where it is possible to apply high voltages 
tothe condenser. It will be apparent from expression (2) 
that if the voltage V, applied to the condenser is in phase 
with the voltage V applied to the field, and a known multiple 
of it, the sensitiveness can be indefinitely increased by making 
V;, large. 

Thus two circular brass plates of 7°3 cm. diameter and 
about 6 mm. apart were tested as a condenser, and found to 
have a capacity of 6x10-® microfarad. This capacity 
caused a deflexion of 180 mm. when 1024 volts were applied 
to the 4000 turn coil, and, by means of a transformer, 
890 volts were applied to the condenser plates through the 
moving coil. Thecapacity tested being so small it was found 
necessary to eliminate capacity effects associated with the 
wires used for the connexions. This was done by taking the 
difference of two deflexions obtained with the connexion to 
one of the plates alternately made and broken. LHarthing 
conditions had to be carefully attended to. Good values 
have been obtained for the specific inductive capacities of 
plates of various dielectrics, but we have as yet not had time 
to properly carry out such tests, which for accurate results 
require balance methods with guard-ring condensers. 

The instrument has not yet been tested with alternate 
currents of higher frequencies than 100 cycles per second, 
but there appears no reason to suppose there will be any 
difficulty in the way of its use for high frequency work. 


Added July, 1910.—Mr. A. Campbell has drawn our 
attention to a paper, previously unknown to us, in which 
Stroud and Oates (Phil. Mag. 1903) describe an instrument 
resembling in some respects the galvanoneter here referred to. 
The paper gives data of the electromagmet showing that it 
contained a greater volume of iron than that of the instru- 
ment here described. But no details are given of the air-gap 
between the poles, and it does not appear that any attempt 
was made to produce a magnetic field whose phase is essenti- 
ally in quadrature with that of the applied voltage. The 
characteristics of the present instrument are the result of 
such a relationship. Jn the Stroud galvanometer the field in 
the gap is probably stronger and less uniform than that of 
the instrument here described ; since the disturbing influence 
of this field on the moving system seems much more serious. 


Pe 820° 


XXVIT. The Production of Cathode Particles by Homogeneous 
Réntgen Radiations, and their Absorption by Hydrogen and 
Air. By R. T. Brarry, M.A, B.H., 1851 Emhabean 
Scholar, Emmanuel College, Cambridge *. 

[Plate V.] 
fi aes properties of the cathode particles produced when 
Roéntgen radiations fall upon various substances have 

been investigated by several physicists. The work of 
Cooksey f and Innes f has shown that the velocities of these 
cathode particles are independent of variations in the in- 
tensity of the Roéntgen radiations used, and also independent 
of the nature of the substance struck by these radiations, 
but that the velocities increase with an increase in the 
penetrating power of the exciting Réntgen radiations. 

The work of Barkla and Sadler upon homogeneous radia- 
tions enables one to use beams of definite quality and differing 
widely in penetrating power, and it seemed that by using 
such beams more precise information might be gained about 
the cathode particles emitted from metals placed in the path 
of such beams. 

In view of the anomalous behaviour of hydrogen with 
regard to ionization phenomena, it was determined to in- 
vestigate the coefficients of absorption by hydrogen and air 
of the cathode particles emitted from a sheet of silver leaf 
which was placed in the path of the homogeneous radiations 
described above. 

The homogeneous radiations from the metals Fe, Ni, Cu, 
Zn, As, Sn, were excited by suitable radiations from a 


Réntgen bulb. 


Fig. 1. 


A homogeneous radiation so produced entered the cylin- 
drical ionization chamber A (fig. 1) through a thin parch- 
ment window. It then passed through a silver leaf, and 

* Communicated by Prof. Sir J. J. 'homson. 


+ C. D. Cooksey, Amer. Jour. Sci. [4] xxiv. 1907, p. 285. 
} P. D. Innes, Roy. Soc. Proc., ser. A. lxxix. pp. 442-462, Aug. 2, 1907. 


mee 
PGA 


Production of Cathode Particles by Réntgen Radiations. 321 


was finally absorbed completely in a thick brass disk EK 
which served as electrode. ‘The cathode particles which 
emerged from H were absorbed by a layer of paper gummed 
on the surface of E, and so contributed nothing to the 
ionization in the region HR. 

Another portion of the radiation, travelling at right angles 
to the plane ot the paper, entered an electroscope (to be 
referred to as the primary electroscope) which served to 
standardize the amount of homogeneous radiation emitted by 
the radiator. ) 

As the quality of the homogeneous radiation is unaffected 
by small variations in the bulb, and as the quantity of radiation 
entering the chamber A is always the same fraction of that 
entering the primary electroscope, no discordance in tlhe 
results can arise from slight variations in the bulb. 


Apparatus. 


A cylindrical brass ionization vessel was constructed with 
an internal diameter of 11 cms. (fig. 2). A circular opening, 


Fig. 2. 


to s06 wel€s. 


€o pump ' 
charcoal Cvbe, 
ey TH, imets, 


(inn 
VELL 
wa) 


to a™ electvoscope 


Mt 
& WLLL coe [2 
| (ie eee ee | 


7 cms. in diameter, was made in the bottom, and a piece of 
copper guuze was fitted into this opening and carefully 
soldered round the edges so as to be quite flush with the 
bottom of the vessel. The details of this arrangement are 
shown in fig. 3. A cap CC was then cast in brass and 


Fig. 3. 


) B Cu gauze B 
2 el 


Farchmen’ 


turned down to fit tightly over the bottom BB. The cap 


was made with a circular opening of the same size as that in 


BB. A sheet of thin parchment was used to cover the Cu 
gauze: its weight was equivalent to that of (005 mm. of Al. 


a22 Mr. R. T. Beatty on the Production of Cathode 


Parchment was found not to be air-tight, and after some 
trials the method of treating it which gave most satisfaction 
was to plunge it in boiling paraffin wax, and then remove 
the excess of wax between filter-papers. This treatment 
increased the weight of the paper to that of ‘01 mm. of Al, 
but still its absorption of even the Fe radiation was quite 
small. The parts shown in fig. 3 were assembled by heating 
BB and CC, covering them with a layer of beeswax and 
resin, placing the parchment in position and pressing CC 
tightly on BB. When the joint was made in this manner 
no trouble with leakages of air ever arose in this part of the 
apparatus. 

A brass ring RR was placed above the gauze, as shown in 
fig. 2, and the opening in it was covered with a parchment 
sheet tightly gummed on. Above this came the electrode H, 
a thick brass disk which conld be raised or lowered by three 
vertical screws passing through ebonite plugs in the disk. 
The points of these screws rested on three small ebonite 
blocks which fitted in recesses in RR (see fig. 2) and a 
conical depression at the centre of each block ensured that 
E could always be replaced in exactly the same position. 
A wire was led from EH to the secondary electroscope, and 
as the vessel was put at a high potential a guard ring 
was used, 

As the vessel had to be opened many times during the 
course of the experiment, it was necessary to have a lid 
which conld be rapidly and efficiently sealed on or removed. 
The usual method of soldering seemed unattractive, and, 
instead, a brass casting AA was screwed on the outside of 
the vessel, forming a deep channel into which melted wax 
could be poured. The lid was then placed in position so 
that the rim fitted into this channel, and a blowpipe flame 
raised the lid to such a temperature (about 100° C.) that 
the wax became fluid. On cooling, the joint became quite 
air-tight. 

The whole vessel was placed above a circular opening in 
a thick lead plate LL, and insulated from the lead by ebonite 
blocks. 

The internal arrangements of the vessel were adjusted as 
follows. A silver leaf was attached to the upper surface of 
RR while the parchment covering of the latter was moist. 
The parchment on drying shrank and formed a perfectly 
plane surface with the leaf adhering to it everywhere. HE was 
then placed on RR and the screws adjusted till the two were 
parallel. A microscope was used to determine the distance 
between these surfaces with a possible error of ‘02 mm. 


Particles by Homogeneous Réntgen Radiations. 323 


This distance was 5 mm. when soft radiations were used. 
RR was then placed in position in the vessel, H was placed 
on top of it and connected to the secondary electroscope. 

The lid was heated and put in position after the screws 
had been connected to earth. 


Method of Experimenting. 


When the bulb was in action the air between E and Rh 
was ionized and a charge communicated to H. ‘The pressure 
of the air inside the vessel was varied, and the ionization in 
the space ER was measured at different pressures (the 
primary electroscope being always used to standardize the 
ionization). 

The sources of this ionization are twofold :— 

(1) Ionization due to Réntgen radiations.alone. This has 
been shown by Crowther* to vary directly as the pressure 
of the air. 

(2) Ionization due to cathode particles emerging from the 
silver leaf. The amount of ionization due to this source will 
remain constant as long as the pressure is great enough to 
absorb all the particles. When the pressure is lower some 
of the particles will reach E before being absorbed, and the 
ionization will decrease. 

Hence, given the actual curve, we can find the part due to 
cathode ionization by drawing through the origin a line 
parallel to the straight portion of the curve, and drawing the 
curve whose ordinates are got by subtracting the ordinates 
of this line from those of the actual curve. 

Fig. 6 (Pl. V.) shows how the ionization due to the 
cathode particles is deduced from the actual curve. 

The pressure at which the ordinate of the cathode curve is 
half the maximum ordinate gives the pressure at which half 
the energy of the cathode particles which start from R 
reaches E. 

Knowing the distance between R and EH, the temperature 
of the room, and the critical pressure, we can now calculate 
the thickness of the layer of air at 760 mm. pressure and 
15° C., which would absorb one half of the energy of the 
cathode particles starting from R. 

Further, we can easily determine from the curves the 
ratio of total ionization due to the cathode particles which 
emerge to the ionization due to Roéntgen radiations in the 
layer of air between R and EH (e. g. PN/QN, fig. 6), and also 
how the total number of ions made by the cathode particles 


* J. A. Crowther, Roy. Soc. Proc. A. Ixxxii. 1909, p. 103. 


324 Mr. R. T. Beatty on the Production of Cathode 


changes when these particles are absorbed completely by air 
and hy drogen respectively. 

For pach radiation a complete curve was first obtained 
with air: then the air was completely pumped out, the final 
exhaustion being effected by means of a charcoal ake sur- 
rounded by liquid air. Hydrogen was then admitted and 
another curve obtained. The Tesults for Fe, Zn, Sn are 


shown in figs. 4, 5, 6 (PI. V.). 
Preparation of the Hydrogen. 


Special care was taken to obtain pure hydrogen, as a very 
small percentage of impurity will increase the ionization in 
it perceptibly. Pure zine and sulphuric acid (Kahlbaum) 
were used, the acid being diluted with ten times its volume 
of distilled water, the hitute being then boiled to expel 
dissolved air: a little copper sulphate was also added. ‘The 
reaction took place in a special form of kipp of small volume, 
the greater part of which was kindly constructed for me by 
Mr. A. Ll. Hughes. The gas was passed through KHO, and 
stored over pure H,SQO,: it was then admitted to the ionization 
vessel through phosphorus pentoxide and three a spiral 
glass tube immersed in liquid air. 

Tables I. and II. show the results obtained with air and 
hydrogen. 

TaBLe J.—Air. 


Radiator. be Ache! C. a, 
Ee see 00804 87 2 149 13-0 
Ci ey, ‘0135 51:9 239 12-4 
i ON ‘0164 42°7 268 11-4 
EOLA ID (0255 27°48 522, 14:3 
Se Be 762 3:97 3-£0 13-9 


TasLe [].—Hydrogen. 


Radiator. be A. R. N. | 
He: pede 0410 L505 5:12 1:01 
ip awie ce a 0733 9°55 5°44 1:00 
Ph he Le 0909 reve! pipe 98 
Saye wae 137 51 T7193 1:00 


2k 


eb 
bo 
Or 


Particles by Homogeneous Réntgen Radiutions. c 
In both tables : 


¢ = thickness of gas in ems. at normal pressure and tem- 
perature required to absorb one half of the energy 
of the cathode particles. 
A = coefficient of absorption of the cathode particles by the 
gas, assuming that they are absorbed exponentially. 


As a matter of fact, reference to the curves shows that the 
exponential law is departed from, but it is convenient to 
calculate X from ¢ for comparison with other values. 

In Table I., C = amount of cathode energy emerging from 
the leaf divided by the ionization in a layer of air "1 om. 
thick just above the leaf (the air being at ‘normal pressure 
and temperature) due to Réntgen radiation. In the pre- 
liminary account * of this research the numbers in this 
column were net reduced to these standard conditions. Also 
it was found that the silver leaf was so thin that with the 
Sn radiation some of the cathode particles produced at the 
back of the leaf were able to penetrate through to the front. 
Obviously as the thickness of the leaf is increased the number 
of cathode particles emerging will increase until all particles 
starting from the back of the leaf are unable to penetrate to 
the front. A thicker leaf was accordingly used, and tlie 
number 8°50 in column C was found. 

Now if N particles emerge per second from a metallic sur- 
face, and if A be their coefficient of absorption in the metal, the 
number of particles produced per second in unit thickness of 
the plate will be NA, if we assume an exponential law of 
absorption. 

X for silver is unknown, but we may assume it to be pro- 
portional to that for air in each caset+. Accordingly the 
numbers in column 4 have been multiplied by those in 
column 3: the numbers in column 5, Table I., have been 
thus obtained. Here T on this reasoning means the total 
cathode energy set free by each homogeneous radiation in 
unit thickness of silver, divided by the ionization in one cm. 
of air just above the silver (and multiplied by an unknown 
constant: the ratio of » by air to » by silver in each case). 

If now we assume that ionization in air is proportional to 
absorption of the radiations by air, and further that the 
absorption in air is proportional to that in silver, then T will 
measure simply the cathode energy set free in silver divided 
by the absorption by silver of the homogeneous radiation. 

The numbers in column 5, Table I., show that the order of 


* Proc. Camb. Phil. Soc. vol. xv. pt. v. p. 416. 
tT Lenard, Wred. Ann. lvi. p. 255 (1895). 


326 Mr. R. T. Beatty on the Production of Cathode 


magnitude is the same in each case. Hence we may assume 
that in the case investigated a constant fraction of the homo- 
geneous radiation is spent in producing cathode particles. 

Further, if the absorption by air of these radiations were 
known, we could find this fraction numerically. 

In Table II., R is the distance which the cathode particles 
travel in hydrogen at normal pressure and temperature before 
becoming half absorbed, divided by the corresponding dis- 
tance in air. 

The ratios increase as the particles become more pene- 
trating. A similar change in R has been observed by 
Lenard * over a much wider range of speeds of cathode 
particles. 

N is the total number of ions produced by a given set of 
cathode particles when totally absorbed in hydrogen, relative 
to the number produced in air under the same conditions. 
The ratio approaches very closely to unity. 

In Table III. are added some data previously found for 
cathode particles by Lenard f and Seitz t. 


TaB_eE ITT. 
Wont Re i Hy; . 
Energy we A for air. d for Hg. cia ch i) Autkority. 
corpuscles in volts. Range in air 
4,000 645 144 4-48 Lenard 
20,000 Si us io Seitz 
30,000 38 ‘47 8:05 Lenard 


It will be seen that the constants relating to the cathode 
particles due to the Sn radiations are very close to those 
found by Lenard for corpuscles possessing a velocity due to 
a drop of potential of 30,000 volts. 

In fig. 8 (Pl. V) the coefficients of absorption of the 
cathode particles are plotted against the absorption by Al of 
the exciting homogeneous radiations. The curves approach 
a linear form. 


* Lenard, Ann. der Phys. xii. p. 7382 (19038). 
+ Lenard, bed. 
t Seitz, Ann. der Phys, xii. p. 860 (1908). 


—~ 
bo 
~l 


Particles by Homogeneous Réntyen Radiations. — + 


Relutive ionization in air and hydrogen due to 
homogeneous radiations. 

The ionization in hydrogen due to soft radiations is so small 
that the straight portions of the hydrogen curves in figs. 4 
and 5 are almost horizontal. A separate set of experiments 
was made to determine this ionization accurately. ‘The 
silver leaf was removed from the ionization vessel and the 
disk electrode replaced by an aluminium wire bent into the 
form of aring. A paper strip was wound round the wire to 
stop cathode radiation from it. The whole volume of the 
vessel was thus utilized. It is not sufficient to fill the vessel 
with air and hydrogen alternately, as tha smali amount of 
cathode ionization from the paper and cardboard lining would 
introduce an effect which would increase the value obtained 
for the direct ionization of the gas. Accordingly the 
ionization was measured at different pressures, as in the 
former experiments, and the ratio of the slopes of the linear 
parts of the curves obtained when the vessel contained air 
and hydrogen successively gave the ratios of the ionization 
in these two gases for each radiation, the gases being supposed 
to be at the same temperature and pressure. 


TABLE LV: 
Radiator. Fe. Cu, Zn. As. Sn. 
--, a Sa q 
EY 1752 | 1745 | 1754 | 1746 | 925-0 


Tonizatien in H, 


The relative ionizations of air and hydrogen by primary 
radiations have been measured by Crowther*. His results 
are given in the following table. 


TaBLE V. (after Crowther). 


Equivalent spark- 
gap of bulb. S| ie ie 16 18 20 24 28 


Ionization in Air 
lonization in Hz“ 


100 | 77 | 476 | 147. | 9°35 | 7-41 | 658 | 5-56 


It will be seen that in Table LV. softer radiations were 
used than could have emerged from the bulb in Crowther’s 


* J. A. Crowther, Roy. Soc. Proc. ser. A. Ixxxii. March 10, 1909. 


rR 


Joc ree 


=e 


Sees 


es a ee 


328 Mr. R. T. Beatty on the Production of Cathode 


experiments. Also the ratios of the ionizations becomes 
constant for such radiations. The value for Sn lies where 
one might expect it considering the penetrating power of the 
Sn radiaion compared to that a the radiation are oma Roéntgen 
bulb with the alternative spark-gaps mentioned. 


Remarks on figs. 4, 5, 6, 7 (Pl. V.). 


In each figure the curves described as “ionization in air” 
and “ionization in H,” are those found directly. The curves 
described as “‘ cathode ionization in air” are obtained by sub- 
tracting the part due to direct ionization by the exciting 

radiation, leaving only that due to cathode particles. The 
pressure at which this curve rises’ to ‘hall its final value is 
indicated by a short horizontal line. 

In the H, curves for Fe and Zn radiations practically all 
the ionization is due to the cathode particles, so that the curve 
found directly also represents the ionization due to cathode 
particles only. 

If we take that portion of the ordinate which is intercepted 
between the “ cathode ionization in air” curve and the hori- 
zontal straight line which that curve ultimately becomes, and 
if we plot the logarithms of these intercepts against the 
corresponding pressures, we get the curves in the figures 
which are described as Jog curves. , The deviation of these 
curves from linearity indicates the departure from the 
exponential law of absorption of the cathode particles. 

However, as the particles are already scattered to a great 
extent before e emerging from the silver leaf, an exponential 
absorption is hardly to be expected. Such an absorption 
would require certain relations between the numbers and 
velocities of particles leaving the leaf at different inclinations 
to the normal. 

In determining the cathode ionization in H, due to Sn 
radiation a special method had to be adopted, as owing to the 
high penetrating power of these particles in H, complete 
absorption of them only took place at pressures nearly atmo- 
spheric. <A series of readings was taken at different pressures 
with and without the silver leaf in the path of the radiation. 
On subtracting corresponding ordinates cf these curves we 
obtain curve IIL. fig. f (Pl. V.) (mar ked with Grosses). The 
curves referr ed to as “ionization in air”? and ‘‘ cathode ioni- 
gation in air” were obtained in the usual way, and the 
abscissee of the latter were multiplied by such a number as 
to make the curve coincide most nearly with the H, curve. 
This number, 7°79, was then taken as the ratio of te pene- 
trating power of the particles in H, to that in air. 


Particles by Hlomogeneous Réntgen Radiations. 329 


Discussion of Results. 


Shortly after the preliminary account of this paper had 
been read *, a research was published by C. A. Sadler f in 
which he studied the emission of cathode particles from various 
metals under the influence of homogeneous radiations. His 
numbers, which agree remarkably well, considering the 
difficulties of the investigation, show that the penetrating 
power of the cathode particle is independent of the metal in 
which it originates, and only depends upon the nature of the 
exciting radiation. The numbers which he obtains fcr the 
coefficients of absorption by air of these cathode particles, 
while trending in the same way as those given in this paper, 
show numerical differences, particularly in the case of the Sn 
radiation. 

Since the particles in Sudler’s experiments emerged from 
the incidence side of the metal, while those treated of in this 
paper came from the emergence side, it seemed possible that 
the cathode particles emerging from opposite sides of the 
plate might show a want of symmetry in penetrating power. 
Accordingly the silver leaf was removed from RR (fig. 2) 
and affixed to the lower side of EH, so that cathode particles 
could only emerge from its incidence side. The coefficients 
of absorption by air were then determined for the particles 
excited by the Sn, As, and Fe radiations in exactly the same 
way as before, and the important discovery was made that 
the coeflicients of absorption of these particles are the same 
whether they come from the incidence or emergence side of 
the silver leaf. 


Tasue VI. 
Radiator. A cn emergence side.| \ on incidence side. 
Bee cee. dbaaeee 3°97 391 
RD ec tial ro wade sead a: 27°43 28:1 
War cltsscls bt deaaca ; 87-2 85°0 


This point being settled, various alterations were made in 
the experimental arrangements in order to test the effect of 
altering the geometrical conditions. It was thought that the 
obliquity of incidence of the radiations (see fig. 1) might 
account for the differences in the values obtained, so the 
se eay, Proc. Camb. Phil. Soe. vol. xy. pt. v. pp. 416-422, February 

+ Sadler, Phil. Mag. Merch 1910, pp. 337-356. 


Phil. Mag. Ser. 6. Vol. 20. No. 116. Aug. 1910. Au} 


330 Production of Cathode Particles by Réinigen Radiations. 


distance between radiator and ionization vessel was increased 
by six ems., but only a change of one per cent. was obtained 
in X when Sn was used as radiator. An attempt was made 
to use a parallel beam of radiation by causing it to pass along 
the axes of a bundle of lead tubes placed between radiator 
and vessel, but the effect then became too small to measure. 

It should be mentioned that the portions of the screws 
between‘ R and E (fig. 2) were covered with paper sheaths 
to prevent the emission of any particles from them. 

An experiment was also made in which a cireular disk of 
lead was placed immediately under the parchment window 
(fig. 2) with a central hole, 2 centimetres in diameter, cut 
out of the lead. 

The Sn radiation was limited by this hole so that none of 
it fell on the screws or on the cylindrical portions of the 
vessel. The value thus found for X was 3°89, nearly the same 
as before. 

Hixperiments are at present being made to find directly the 
amount of corpuscular energy produced i in a substance when 
a definite quantity of Réntgen radiation is absorbed by it. 
The results of these experiments can be applied to the re- 
determination of most of the numbers given in this paper by 
an entirely different method. The author hopes to clear up 
in this way the differences already mentioned. 


Summary. 

(1) The absorptions by hydrogen and air of cathode 
particles excited by homogeneous Rontgen radiations have 
been measured. 

(2) It has been deduced from the numbers obtained that 
the amount of corpuscular energy set free in silver is roughly 
proportional to the energy of the exciting radiation absorbed 
by the silver. 

(3) The relation between the absorption of the cathode 
particles by air and the absorption of the exciting homo- 
geneous radiations by aluminium is nearly linear. 

(4) The direct ionization in hydrogen relative to that in air 
has been measured when homogeneous radiations of different 
penetrating powers were used to cause the ionization. 

(5) When a given set of cathode particles spends all its 
energy in producing ions in air or in hydrogen the total 
number of ions produced is the same in each gas. 

(6) The penetrating power of the cathode particles has 
been found to be the same whether emitted from the ye yo 
or incident side of the leaf. 


I wish to thank Professor Sir J. J. Thomson for his interest 
in these experiments. 


east 


XXVIII. A Difference in the Photoelectric Eject caused by 
Incident and Emergent Light. By Orro STUHLMANN, Jr., 
A.B., Experimental Science Fellow, Princeton Univer sity®. 


Introduction. 


ECENT investigations have shown that the ionization 

produced by the secondar y rays arising from a thin metal 

plate traversed normally by a primary beam of y, Rontgen, 

or 8 rays, is greater on the emergent than on the incident 
side. 

W. H. Bragg t, in his work on the nature of the y-rays 
and Réntgen rays, showed that if y-rays pass through a thin 
plate so that the absorption is negligible, the amount of 
emergent radiation is greater than the incident. This re- 
markable want of symmetry he points to as appearing fatal 
to the ether pulse theory of y-rays and, from their many 
points of similarity, of the Roéntgen rays also. In a later 
paper on the nature of y-rays W. H. Bragg and J. P. V. 
Madsen ft also show that this want of symmetry holds for both 
y and B-rays. This lack of symmetry for secondary Rontgen 
rays was also discovered by W. H. Bragg and J. L. Glasson§. 
They showed that this want of symmetry was in general 
more pronounced for the softer rays than for hard rays. 
That the proportion of emergent to incident radiation differed 
considerably for the different radiators, but was much the 
same for different thicknesses of screen, except that the pro- 
portion tended to increase slightly as the screen was made 
thicker ; and the tendency was most pronounced in the case 
of those metals which gave out a quantity of soft secondary 
radiation, the emergent secondary rays being generally in 
excess of the incident. 

The present experiments were made to see if there was 
any ditference in the photoelectric effect caused by the 
incident light and the light which emerged after passage 
through a thin metal film. 


' Apparatus. 


Thin films of platinum were prepared by sputtering in 
vacuo from a platinum cathode on to quartz plates 1 mm. 
thick. Two plates were sputtered simultaneously so as to 
insure the same thickness for both films. These were now 
mounted in the centre of two similar brass cylinders A and B 
Communicated by Prof. O. W. Richardson. 

Bragy, ‘ Nature,’ Ixxvii. pp. 270-271, Jan. 23, 1908. 

Bragg and Madsen, Phil. Mag. xvi. pp. 918-939, Dec. 1908. 

§ Brage and Glasson, Phil. Mag. xvii. pp. 855-864, June 1909. 
Li2 


ttot & 


Se eee eee —_ a 
2 = = mPa a Scie A 


332 Mr. Otto Stuhlmann on a Difference in Photoelectric 


(fig. 1), so that their planes were perpendicular to the axes 

of the cylinders. The plates had an area of one square centi- 

metre, with parallel faces, although not ground optically 

accurate. ‘The corners were ground round to prevent leakage 
i Fig. 1. 

hi-200 Volts [1]-—> Earth. 


to the cylinders. They were mounted in brass clamps sup- 
ported through the top of an earthed metal chamber, C, by 
means of hard rubber bushings. A pointer, D, attached 
to the supports, with suitably arranged stops allowed them to 
be turned around their axes. The plates were connected 
to the negative terminal of a 200 volt battery, the other ter- 
minal of which was grounded. Hxperiments showed that 
this was sufficient to produce saturation. 

The brass cylinders, 5 cm. long and 2°5 cm. in diameter, 
were provided with caps the opening through which was 
1:5 cm. in diameter. This insured against the possibility of 
having photoelectric ions drawn over into the adjoining 
cylinder. The cylinders, resting upon sealing-wax supports, 
were connected with a Dolezalek electrometer, giving about 
900 divisions deflexion per volt, which was placed in an 
adjoining metal case connected to earth. A narrow beam of 
ultra-violet light, stopped down to 2 mm. by the opening at 
S, was allowed to pass down the axes of the two cylinders 
normally to the plates. An are, with both rods made of soft 
iron, running on 4 amp. direct current, was used as the source 
of ultra-violet light. It was enclosed in a light-tight box 
provided with a shutter, by which the beam of light passing 
through the apparatus was controlled. The distance of the 
arc from the plate B was 55 cm., and from the plate A 60 5 em. 


Effect caused by Incident and Emergent Light. 333 
i 1 g € 


Method of Experimenting. 

The saturation current from the illuminated plates to the 
cylinders could be measured by the following differential 
method. ‘The keys G, H were opened simultaneously. A 
beam of light was then thrown on the plates until a 
measurable deflexion of the electrometer was attained. This 
deflexion was equal to the difference in the saturation currents 
from A and B. A was then grounded by connecting G to 
earth ; the resulting deflexion giving the saturation current 
from B. The difference between these two readings gave 
the corresponding reading for A. 

The experiment consisted in measuring for various thick- 
nesses of metal films, the saturation current for two successive 
positions of the plates (fig. 2). 

Position (1), A and B so placed that their film side faced 
the light. 2 

Fig. 2. 


A B A B 


Direction 


Q 
Light, 


POSITION (1) POS\TION (4) 


Pesition (2), A with the film side towards the light, and 
B with the film side away from the light. 

Readings were taken alternately for Positions (1) and (2). 
Thus in every experiment two similar plates with equal 
thickness of metal films were used, and the ratios of the satu- 
ration currents A/B for Positions (1) and (2) successively 
determined. Hence it is seen that the plate B is always 
compared for each of the two Positions with plate A. Thus by 
always referring the measurements to A, the standard plate, 
the otherwise troublesome variations of intensity of the arc 
were rendered harmless. Unless the films were very thick 
it was always found that A/B for Position (1) gave rise to a 
relatively greater photoelectric effect than the ratio A/B in 
Position (2), although in the latter case the emergent beam 
of light was obliged to pass through the quartz plate before 
affecting the film. 

The object of the investigation is the comparison of the 
ionization due to incident and emergent beams of light of 
the same intensity. A legitimate way of measuring this 
intensity is by means of the photoelectric effect such beams 


534 Mr. Otto Stuhlmann on a Difference in Photoelectric 


produce. Different observers have shown that beams of 
similar composition under similar conditions produce a photo- 
electric effect proportional to the intensity of the light, unless 
the intensity is very small. 

If there were no difference in the photoelectric effect pro- 
duced by incident and emergent beams of the same intensity, 
the value of the ratio A/B for Position (1) should always be 
less than that of A/B in Position (2), on account of the 
absorption by the quartz plate and by the film. As a matter 
of fact the reverse was always found to be the case with thin 
films, showing that the ionization is greater for an emergent 
than an incident beam of the same intensity. It is not 
dificult to allow for the absorption by the quartz plate, as 
will be shown below, but the absorption by the film is a more 
serious matter. This arises from the fact that these films 
are so thin that the electrons must be regarded as produced 
throughout the volume of the film. The method was there- 
fore adopted of seeing if the value of the ratio of the fractions 
A/B in the two positions did not approach a limiting value, 
as the thickness of both films was indefinitely diminished. 
This was found to be the case, as will be seen in the sequel. 
Under these circumstances one could be certain that the 
limiting value corresponded to the case in which the absorp- 
tion of the light by the film was negligible ; so that, after 
allowing for the absorption by the quartz plate, the only 
difference between the ratio A/B in the two positions in the 
limiting ease will arise from a difference in the ionization 
produced by incident and emergent beams of equal intensity. 

The following table gives a set of readings characteristic 


Platinum Set 20. 


Position (1). Position (2). 


meee.) |) ar | dae edad ae Sa 


20-0 76:0 56:0 | *736 ies ate see eo 
27-0 99°3 72:3 | °726 39°0 1150 76:0 661 
25°0 95°0 19-0 4 shod 25°0 70°5 45°5 647 
31:0 1150 82°04 aot 31-0 88°5 575 649 
35°0 1400 |105:0| -750 35°2 102°8 67°6 ‘657 
27-0 99:0 fC UE lel 2, 260 74:0 48:0 649 
42°0 1730 {1360 | °763 42°0 118-0 76°0 "644 
30°0 117°5 87:5 | °744 380 108-0 70:0 ‘649 


Average “739 Average *651 


a) 
Bano (oy =1:13 


Effect caused by Incident and Emergent Light. 335 


of the experiments on very thin films of platinum. The 
maximum deviation from the mean is about 1°5 per cent. 

By changing the thickness of the films the following values 
were obtained:— 


Platinum. 
rs) A 
AL IED NEES Un 
i | | | — Ratio (2) og 4 Position (2). 
oO 
A A a, is 
(2)= a° (= 5 Equal to log its 
a 590 398 “674 1399 
b 652 726 1:09 18 
c 651 ‘739 1:13 ‘131 | 
d 587 617 1:05 209 
e 647 ‘708 1:09 149 
f 573 656 1-14 183 
g 531 552 1-04 "257 
h 5384 536 1:00 “270 
i 624 ‘694. Pit "158 
7 632 ‘657 1-04 "182 
i 632 ‘708 1°12 "148 
l 644 *730 1-12 "136 
1 534 "536 1:00 ee 
2 628 "696 1-10 "156 
3 635 ‘679 1-07 168 
4 639 ‘719 112 "143 
5 622 677 1-10 "169 
6 549 578 1:05 Dae. 
7 550 475 "863 "320 
8 "536 566 1:05 "247 
9 ata "496 ‘910 304 
10 ‘611 397 649 “400 
kL *660 394 “G00 404 
13 561 "590 1-05 *228 
16 "D82 "685 Bi °164 
20 "OLS 694 LLS "158 
29 “866 ‘966 EL “015 
24 666 "754 L138 “122 


From the above data we see that the effect is a function of 
the thickness. It was found impossible to determine the 
thickness of these very thin films directly. From the ratio 
of A/B in Position (1) it is possible, however, to compare 
their relative thicknesses with accuracy. For it is known 
that when light traverses an absorbing medium the logarithm 
of the intensity varies as the thickness of the medium traversed. 
Hence from the preceding arguments it will be seen that values 
of log B/A Position (1) (equal to log ip/?) will be proportional 
to the thickness of the film plus that of the quartz plate used. 


836 Mr. Otto Stuhlmann on a Difference in Photoelectric 


Hence a curve (see fig. 3) between the ratio of emergent to 
incident effect plotted against log io/2 will be identical in form 
with that plotted against thickness. : 


APS} Wwopiou: 03 yusBisuis fo oljoy * 


It is seen from the curve (fig. 3) that as the metal decreases 


Effect caused by Incident and Emergent Light. 337 


in thickness the emergent beam gradually becomes more 
predominant in its effect, until a certain thickness 1s reached 
where the ratio of emergent to incident light attains a constant 
value 1:12 to 1:0. The ratio remaining constant over so 
large a range of thicknesses for the thin films of metal, shows 
conclusively that the absorption of light and electrons by 
these films as the beam of energy passes through them is 
negligible, and falls within the experimental error. So that 
the intensity of the incident and emergent beams must be 
sensibly equal, and hence the value arrived at above must 
be a true difference between the incident and emergent 
light. 


Relative Absorption of Quartz Plate. 


In order to determine how much of the emergent beam 
was absorbed by the 1 mm. quartz plates, upon which the 
films were mounted, the following method was resorted to. 
A relatively thick film was sputtered on a 1 mm. quartz 
plate. A thick film was preferably used because less error 
was involved in the measurement of its photoelectric effect, 
since for equal intensities of light relatively thicker films 
gave a larger photoelectric effect per unit time of exposure 
to the light. With this film the experiments for Position (1) 
and Position (2) in fig. 2 were repeated, and its ratio of 
emergent to incident effect noted. 

A second blank quartz plate 1 mm. thick was now rigidly 
fastened to the blank side of the sputtered plate B. So that 
now the light had to pass through 2 mm. of quartz when the 
above readings were repeated. This was repeated for 3 and 
4 mm. quartz plates by addition of a 2 and 3 mm. plate to 
the original 1 mm. sputtered plate B. 

The results were then plotted as shown in fig. 4 (p. 338). It 
is seen that for a value of emergent to incident effect for the 
1 mm. plate equal to 1:10,a value 1:15 should have been 
attained if no quartz had been present. So that plotting the 
eurve for 1 mm. quartz equal to 1:12 for very thin films, 
parallel to the original curve, we get a value for the ratio of 
the emergent to the incident beam equal to the ratio 1:17 to 
1-0, or an increase of 17 per cent. 

The above results have been confirmed by reversing the 
direction of the light. For this purpose a speculum metal 
mirror was placed so that the reflected beam of light could 
be alternately sent through the cylinders in opposite direc- 
tions without changing the position of the plates. Experi- 
ments were also made to determine whether a slight deviation 


338 Mr. Otto Stuhlmann on a Difference in Photoelectric 


from the normal, on the part of the plates, would affect the’ 
results. A change of five degrees in the position of the 


> 


tf Quartz wi mm. 


Thickness o 


98 _ 1.06 
Ratio of Emergent to Incident Eypect. 


plates showed no measurable effect. Blank quartz plates 
placed in the cylinders gave negative results. A test for 
photoelectric effects arising from scattered light gave no 
measurable effects, since the aim in the design of the appa- 
ratus was to use so small a beam that it cleared all openings 
in the apparatus and the reflected beams passed out, back 
through the openings, through which the incident beam 
entered. 

An attempt was made to estimate the absolute value 
of the thickness of the thin films used, by measuring 
their resistance. For this purpose “Set 2” of the above 
data was used. In order to get proper electrical connexions 
the two opposite edges of the film were silvered by means of 
the Rochelle Salt Method. Thus a strip of metal film ‘78 cm. 
long and 1 cm. wide was left exposed. Fine copper wire 
was wound around these edges and electroplated, by means 
of a copper deposit, to the silver below—thus furnishing a 
good connexion between the copper wire and the film of 


Effect caused by Incident and Emergent Light. 339 


platinum. This was now placed in one arm of a Wheatstone’s 
bridge and its resistance measured. Using a value of the 
specific resistance for thin platinum films equal to 5 x 10-° ohm 
as found by Patterson*, a value for the thickness was found 
comparable to'3x10-cm. Hence the thinnest films used 
were evidently of order 10-7 cm. in thickness. Similar values 
were found by Patterson for the thinnest films measured 
by him. 

Experiments on other metals are now in progress which 
will determine the variation of this forward effect with the 
atomic weight of the element. 


Conclusion. 


The above experiments show that when beams of ultra- 
violet light of equal intensity are compared, the ionization 
they produce is greater on the emergent than on the incident 
side of a thin platinum film. Tor a film so thin that the 
absorption of the light in it is negligible, the ratio of the 
ionization on the emergent to that on the incident side is as 
1-17 is to unity. There is thus an increase of 17 per cent. 
in favour of the emergent side of the film. 

An effect of this kind would obviously be expected on any 
corpuscular theory of light. It can also be explained on an 
undulatory theory by a process of the nature of light-pressure, 
which tends to push the electrons forward in the direction 
in which the light is propagated. In its ordinary electro- 
magnetic form, however, the undulatory theory does not 
appear to give rise to effects large enough to explain the 
phenomena observed. ‘The difficulty appears to be similar 
to that which arises when the ordinary theory attempts to 
explain why the ultra-violet light is capable of causing the 
expulsion of the electrons, with their observed properties, 
under any circumstances. 


-. This Investigation was suggested by Professor O. W. 
Richardson, and I wish to thank him here for valuable 
suggestions and advice throughout the course of the work. 


Palmer Physical Laboratory, 
Princeton University, 


* J, Patterson, Phil. Mag. iv. p. 663 (1902). 


— = 


PRS EOO NG 


XXIX. The Relation between Uranium and Radium.—V. 
By FREDERICK Soppy, M.A., /.B.S. 


a the last paper on this subject (IV., Phil. Mag. Dec. 

1909, p. 846) details were given of the measurements 
of the quantities of radium in the three uranium solutions, 
purified by Mr. T. D. Mackenzie some years ago. It was 
shown that, within the supposed error of measurement, the 
growth of radium was proceeding ata rate proportional to 
the square of the time, and the period of the long-lived 
intermediate parent of radium (ionium) was deduced from 
this to be 18,500 years in the case of one solution and 26,000 
years in that of another. Subsequent measurements have 
not confirmed these conclusions, and the former apparent 
rate of growth of radium according to the square of the time 
has not been maintained in any of the three solutions. A 
recent redetermination of the constant of the instrument with 
two of the former standards has shown that it has changed 
appreciably, the electroscope having become apparently more 
sensitive. The ‘‘constant” + is now 5:2, whereas the last 
calibration tests, carried out over a period between 350 and 
250 days previously, had given the mean value of 5°78. 
Such a change was not anticipated, as neither the electro- 
scope nor its reading microscope has been in any way moved 
or altered in adjustment since first set up, and both have 
been kept exclusively for these measurements. 

It is very difficult to get a method of checking the sensi- 
tiveness of an instrument of this character accurately, which 
shall be quite free from uncertainty. It is necessary to use 
a standard solution containing a minute amount of radium, 
and, as is well known (Mme. Curie, Le Radium, 1910, vii. 
p- 65), these tend to change with time, the amount of radium 
apparently diminishing, very possibly through the solution 
of a precipitating constituent from the glass. For fear of 
this I made some of my standards with very minute quantities 
of radium (as low as 5x 10-" gram). It is unlikely, how- 
ever, that the quantity of radium could increase, as it would 
have to do in this case to explain the change of the constant 
of the instrument. In the same paper, however, Mme. Curie 
called attention to variations amounting to 5 per cent. or 
more in the estimation of radium by means of the emanation 
when periods of accumulation longer than 48 hours are 
employed. Some other indications of a possible ‘‘ radium X ” 

* Communicated by the Author. 


+ The constant is the number of units of radium (10-1? gram) required 
to produce unit leak, 7. e. one division a minute. 


Relation between Uranium and Radium. 341 


intermediate between radium and its emanation were also 
obtained. - A freshly crystallized radium chloride appears to 
suffer a slow augmentation in the rate of production of the 
radium emanation, whereas a solution from which the radium 
had been precipitated by sulphuric acid gave a progressively 
diminishing rate. Still more recently Hahn and Meitner 
(Phys. Zeit. 1910, xi. p. 493) have observed a continuous 
increase in the activity of a radium salt for many weeks after 
the emanation has reached equilibrium, to a value more than 
twice as great as the original equilibrium value, which they 
regard as indicative of the existence of a radium X. So that 
it is possible that the apparent variation of the sensitiveness 
of the electroscope may admit of another explanation. 

However this may be, in view of the results obtained 
during the past year with the uranium solutions, it is necessary 
to withdraw the estimates of the period of the intermediate 
parent of radium given in the last paper, and to treat the 
results as affording data for the calculations of the minimum 
period, but not as furnishing satistactory evidence so far of 
the production of radium from uranium. I do not think 
there is any doubt that a steady increase in the amount of 
radium in all three solutions is taking place, but, as pointed 
out in a previous paper, this may be explained by the possible 
presence from the start of a minute quantity of the long-lived 
parent of radium which I have shown is present in com- 
mercial uranium salts in quite easily detectable amount. 

With regard to the minimum period the solution containing 
408 grams of uranium purified on 13/12/06 gives the most 
definite results on this point. When last tested, three and a 
half years from purification, it contained between 8 and 9 
units of radium. Seven good measurements have been made 
on this solution in the last two years, some of which were 
detailed in the last paper. Even ifit be assumed that initially 
both radium and its parent were completely absent, which is 
an unlikely supposition, and that the radium present now has 
all actually been formed from the uranium in the three and 
a half years, the period of the long-lived intermediate body, 
supposing there is but one, must be between 37,500 and 
33,300 years. ‘There is thus little doubt that the estimates 
in the last paper were too low, and that there are not as yet 
sufficient data from which to set an upper limit to this period. 
There are many reasons for thinking it may even be many 
times the minimum period assigned, and some of these 
reasons are discussed in the next two papers. 

It thus becomes a matter of considerable importance in the 
present state of radioactivity to fix at least an upper limit 


342 Mr. F. Soddy on the 


for the period. This could readily be done with the prepara- 
tion of thorium oxide described by Stefan Meyer and Egon 
v. Schweidler (Wiener Anczeiger, Sitzung. 11/6/09). which 
has been separated by Auer v. Welsbach ‘from the products 
obtained by Haidinger and Ulrich (Weener Ber. 1908, cexvii. 
p- 621) from the radium residues of 30 tons of Joachimsthal 
pitchblende, and which is stated to contain about 0°25 per 
cent. of ionium. ‘This, however, appears to have been 
obtained from measurements of the a-ray activity, by 
assuming for the substance a period the same as for radium, 
whereas the period and also the percentage of ionium is at 
least fifteen times greater. A maximum limit to the amount 
of ionium present could be fixed by subtracting from its total 
mass the mass of thorium oxide present. The latter could 
be determined by comparing the amount of thorium emana- 
tion generated in a solution, preferably boiling, of the 
substance with that from a standard thorium solution of the 
same age since preparation under the same eonditions 
(Rutherford and Noddy, Phil. Mag. 1902, 1 lv. p. 378). The 
maximum period of ionium required is simply given by 
dividing the maximum limit that can be assigned to the 
mass of the ionium by the mass of radium generated by 
the preparation in one year. This would fix the period 
between two limits, and however wide apart they proved at 
first to be, a very valuable step would have been made. 


Physical Chemistry Laboratory, 
University of Glasgow. 


XXX. The Rays and Product of Uranium X.—II. 
By FREDERICK Soppy, M.A., F.R.S.* 


A ar conclusions in the precéding paper as to the minimum 
period of ionium modify essentially the deductions pre- 
viously drawn from the behaviour of the uranium X prepa- 
rations separated from 50 kilograms of uranium nitrate 
(Phil. Mag. 1909, xviii. p. 858), and it is desirable to give 
some account of the further progress of this work. It was 
pointed out that the failure to observe the growth of an 
a-radiation during the decay of these preparations was 
inconsistent with the measurements of the rate of production 
of radium from uranium if uranium X were really in the 
uranium—radium series. As, however, it has just been 
shown that the period of ionium was probably greatly under- 
estimated, this failure rather affords additional evidence 


confirming the extremely long period of this body. Using 
* Communicated by the Author. 


Rays and Product of Uranium X. 343 


the new minimum value of 35,000 years instead of the old 
value of 18,000 years for the period, it follows that the 
maximum growth of a-rays to be expected from the uranium X 
preparations corresponds to only 5 divisions a minute, and 
the detection of this would have been doubtful. Whereas, if, 
as is not improbable, the real period is much longer than this, 
the theoretical growth of a-rays becomes certainly too small 
to be detectable. | 

In all the preparations the @-rays have decayed normally, 
so far as can be seen, and in the older preparations are now 
scarcely detectable. For all the preparations, measurements 
of the a-radiation by the method and apparatus previously 
described can now be done without exciting the magnet, 
and practically the same results are now obtained for the 
a-radiation, whether the magnet is excited or not. This 
shows that the method is trustworthy, and that the small 
proportion of 8-rays escaping deviation did not interfere with 
the initial measurements of the feeble e-radiation, when the 
preparations were intensely active. This proportion can now 
be accurately calculated from tne known constant of decay, 
00282 (day)-! (Soddy and Russell, Phil. Mag. 1910, xix. 
p- 847). About 7}, part of the @-rays escape deviation, 
and therefore it would appear must possess a value for Hp 
above 8640. 

Most of the preparations have now decaved sufficiently far 
for the e-radiation to be accurately measured in an ordinary 
electroscope. There certainly has been no increase in the 
a-rays. Indeed, the results rather indicate a very slow decay; 
but more time must elapse before this can be verified. 

With regard to the a-ray measurements in hydrogen with 
the magnet, the results with all the preparations show practical 
constancy from the start, and the small variations are doubtless 
due to unavoidable changes in the atmospheric conditions at 
the time of measurement. The following table refers to the 
observations with the preparation of the fourth separation. 
The first 10 observations previously given (bed. p. 863) varied 
between 37°1 and 40°5, with 38°9 as the mean. 


| Time (days) ... 61 


0-33 
(Mean of 10) 


| 
Or 
—" 
(we) 
(ee) 
i} 

~ 
(@ 6) 
Hee 


The preparations of the second and third series have 
behaved quite similarly. The oldest of these was prepared 


— 


a egg er a a 


344 The Rays and Product of Uranum X. 


nearly 18 months ago. Any change has been in the nature 
of a slight decay rather than an increase. 

Measured in an ordinary cylindrical electroscope (13 cm. 
high by 10°5 cm. diameter), the a-activity of the more active 
of the preparations, namely, those first prepared and therefore 
containing most of the initial impurities in the uranium, are 
comparable with that of a similar surface (10 sq. cm.) of 
uranium oxide. The least active preparations (those of the 
third separation) possess only about a third of this activity. 
For the latter it can easily be calculated that the permanent 
a-radiation produces only about gg!55 part of the leak which 
would have been produced in the same electroscope initially 
by the @-radiation. The calculated initial @-activity of each 
of the more active preparations corresponds to about half a 
million scale-divisions per minute. That produced by the 
a-rays per sq. cm. of uranium oxide surface is about 44. 

Other tests have been done to detect a possible growth of 
actinium in the preparations. The method employed consists 
in placing closely over the (positively charged) platinum 
tray containing the preparation a negatively charged brass 
plate, removing the latter after a definite period, and mea- 
suring its e-activity and its rate of decay. Hvidence of 
actinium was first found when the first preparation had 
decayed far enough to allow of measurements in an ordinary 
electroscope. The readings steadily increased after inserting 
the preparation due to an emanation being generated. The 
active deposit test showed that actinium was undoubtedly 
present. The activity of the active deposit decayed to halt- 
value in 37°5 minutes, with an initial delay characteristic of 
actinium A. After two hours exposure the activity obtained 
initially increased the natural leak of the electroscope about 
10 times, and corresponded to the specific a-activity of 
0°65 sq. cm. of uranium oxide. ‘The second preparation 
gave a detectable active deposit, but 1t was only one-fifteenth 
as much as the first. In the single preparations of the third 
and fourth separations, no actinium whatever could be 
detected. Quite recently, a combined test with all four pre- 
parations of the third and fourth separations showed a just 
detectable active deposit. Naturally these observations will 
be continued. Some of them have been in progress seven 
months, and have shown that if a growth of actinium occurs 
at all it must be very slow. Its presence in the first prepa- 
rations may therefore be ascribed to initial impurities in the 
uranium, which are rapidly separated by the successive 
erystallizations. 

The general result of this investigation is to show that 


Ratio between Uranium and Radium in Minerals. 345 


uranium X loses its activity completely without the formation 
of any product possessing a detectable radioactivity. We 
must therefore suppose either (1) that uranium X is not in 
the uranium-radium series, (2) that long-lived new rayless 
products must exist in the series, or (3) that if uranium X 
changes directly into ionium the period of the latter must be 
greater than 30,000 years. There is good reason to believe 
that the last explanation may prove correct; so that to achieve 
the object for which the investigations were commenced still 
larger quantities of uranium must be dealt with, or the 
methods must in some way be greatly improved. 


Physical Chemistry Laboratory, 
University of Glasgow. 


XXXI. The Ratio between Uranium and Radium in Minerals. 
By Freverick Soppy, MW.A., &.R.S., and Ruta Prirrer, 
B.Sc.* 


Mite. Guepirscw (Compt. Rend. 1909, exlviil. p. 1451 ; 
exlix. p. 267), in a reexamination of the ratio of uranium 
and radium in minerals, found small but distinct variations 
in the ratio for Ceylon thorianite, Joachimsthal pitchblende, 
and French autunite. Thorianite being probably a very old 
mineral and autunite a very recent one, the results sug- 
gested that the older the mineral the greater the ratio of 
radium to uranium. Neither Boltwood nor Strutt in their 
original determinations of this ratio refer to autunite, but 
the latter included thorianite (Phil. Mag. 1905, ix. p. 599; 
Proc. Roy. Soc. 1905, A. Ixxvi. p. 88). Mlle. Gleditsch first 
separated the radium chemically from the minerals before 
estimating it; and it seemed very desirable to repeat the 
work, determining the radium in the usual way, without 
carrying out this separation. This has been done for speci- 
mens of the three minerals mentioned, the radium having 
been estimated by comparison of the amount of radium 
emanation generated, after periods of accumulation of a 
month or longer, in solutions of the minerals containing a 
fraction of a milligram of uranium preserved in sealed 
flasks. For Joachimsthal pitchblende we employed standards 
I. and V., previously used for calibrating the electroscope, 
and described by one of us in an earlier paper (Phil. Mag. 
1909, xvill. p. 849), where a full description of the method 
of estimating the radium is given. So far attention has 
been mainly directed to thorianite ; for, as Mlle. Gleditsch 
* Communicated by the Authors. 
Phil. Mag. 8. 6. Vol. 20. No. 116. Aug. 1910. va A 


ie 


So ee. 


eee 


346 Mr. F. Soddy and Miss Ruth Pirret on the 


points out, it is more difficult to explain why a mineral 
should possess a greater than normal ratio of radium to 
uranium than the reverse. But the results with this mineral 
are not yet decisive. However, the results with autunite are 
of interest in connexion with the subject discussed in the 
preceding paper, so that it seems advisable to give an account 
of what has so far been done. The thorianite solution for 
the radium estimations was made by dissolving sixty milli- 
grams of the powdered mineral in nitric acid, which had 
been distilled over barium chloride, diluting without filtering 
to a known weight and sealing up a known fraction, deter- 
mined by weighing. The specimen of thorianite used left 
about 3°5 per cent. of residue insoluble in nitric acid. 
Previous experiments had shown that this residue, when 
examined in an electroscope, was practically free from radio- 
active matter. No §-activity whatever could be detected in 
the insoluble residues from 16 grams of the thorianite, used 
in the analyses, which had been sealed up in test-tubes for 
four months. Therefore neither uranium nor radium could 
have been present in appreciable amount. 

The autunite employed was of Portuguese origin and 
consisted of beautiful crystals of the pure mineral in a 
clayey matrix. Twenty grams were used, and the whole of 
the autunite, consisting of about 40 per cent. of the total 
mineral, was dissolved out in hydrochloric acid. The residue 
was sealed up in a test-tube. After a month no f-activity 
whatever could be detected from the tube, showing that the 
whole of the uranium and radium had been dissolved. In 
the hydrochloric acid solution the uranium was estimated 
in one part and the radium in another. | 

Estimation of Uraniumin Thorianite.—The method mainly 
employed was based upon the one described by Boltwood 
(Am. J. Sci. 1908, xxv. p. 269), and used by him in esti- 
mating the amount of uranium in many minerals (including 
thorianite) containing uranium and thorium. We found 
however that, as described, it was impracticable for the 
following reason. The solution of the mineral in nitric 
acid after removal of the lead with sulphuretted hydrogen 
is evaporated to dryness at 110°. In Boltwood’s description 
the uranium is mainly removed from the dry residue by 
extraction with pure dry ether. The part insoluble in ether 
is then dissolved in dilute nitric acid, treated with oxalic 
acid to remove thorium and other rare earths. After removal 
of oxalic acid by ignition the dissolved residue is treated 
with ammonium carbonate and sulphide to remove iron &e. 
and the uranium, recovered from the filtrate by boiling, is 


Ratio between Uranium and Radium in Minerals. 347 


joined to that from the ether extraction. However, we 
found that the ether extract contains actually more thorium 
than uranium. This was proved for the aqueous solution of 
the evaporated ether extract, both by precipitating with 
oxalic acid and by tests on the amount of thorium emanation 
generated. So the thorium and other rare earths were pre- 
cipitated with oxalic acid from both the ether extract and 
residue. But it was found that the extraction was quite 
unnecessary, the same results being obtained whether it was 
employed or not. The uranium finally in some cases was 
precipitated by ammonium sulphide, and weighed as U;0, 
after ignition in oxygen; in others, as phosphate by micro- 
cosmic salt and sodium thiosulphate (Brearley, Analytical 
Chemistry of Uraniun, p. 7). 

Five analyses were performed of the uranium which gave 
results comprised between 19°74 and 20°56 per cent. of 
uranium. The mean, which agreed well with the three most 
satisfactory determinations, was 20:0 per cent. 

For autunite the uranium was precipitated as phosphate 
by microcosmic salt and sodium thiosulphate in presence of 
free acetic acid. 

Ratio of Radium to Uranium.—The results are given in 
the following table. In the first column is given the name 
of the mineral, in the second the quantity of uranium in the 
solution employed, in the third the leak of the electroscope 
in divisions per minute, and in the last the leak per milli- 
gram of uranium. If the ratio of uranium to radium is the 
same for all minerals, the figures in the last column should 
be constant. 


Mineral. (milligrams). | Leak. cee 
Bride. foe = BAe 
Miterianite © cesses... 0613 aa i om \ 67-4 
PPREHULG | osc cscenduns 0-834 24:3 29°1 a 


Thus the ratios for thorianite and pitchblende come out 
very nearly the same, the thorianite being about 3 per cent. 
higher than pitchblende; while for autunite the ratio is 
very low, being less than half (44:5 per cent.) of that of 
pitchblende. 

The measurements, therefore, so far as they have gone 
bear out those of Mlle. Gleditsch, in that they show the ratio 

| 2A 2 


348 Mr. F. Soddy and Miss Ruth Pirret on the 


is not constant in the three minerals. But pitchblende and 
thorianite are so near that the difference can scarcely be 
regarded as significant. It must be remembered that the 
figures for pitchblende depend upon the accuracy of the 
uranium estimation in the standards prepared previous to 
this work, for the purpose of calibrating the electroscope. 
There is no reason to doubt the accuracy of these analyses, 
but the results call for further measurements with other 
samples of both minerals before the matter can be considered 
settled. The specimen of thorianite selected was the purest,-— 
that is to say, the one containing the least foreign matter 
insoluble in nitric acid,—of many which have been examined. 
But its uranium content is unusually high, and the specimen 
in this respect cannot be considered representative. One 
curious circumstance should be recorded. In the paper 
immediately preceding this it is shown that the constant of 
the electroscope has changed apparently since these pitch- 
blende standards were first prepared. ‘This change was 
actually discovered in the course of the present work. The 
ratio of radium to uranium in thorianite appeared, using 
the original value of the constant of the instrument, to be 
about 18 per cent. higher than for pitchblende, which is 
almost exactly what Mlle. Gleditsch found. Buta redeter- 
mination with the old standards brought to light the change 
of sensitiveness of the electroscope ; and although the cor- 
rected ratio is still higher for thorianite than for pitchblende, 
the difference is so small that it cannot be accepted without 
further confirmation. The point at issue is rather a fine 
one, and calls for a degree of accuracy not easily obtained in 
such measurements. 

As regards autunite, however, the variation in the ratio 
for the specimen examined (of Portuguese origin) is far 
greater than that found by Mlle. Gleditsch for French 
autunite. The Portuguese autunite has little more than half 
the radium in the French autunite. The natural explanation 
of these results is that the mineral is of so recent formation 
that the uranium-radium series is not yet in equilibrium. 
In the preceding paper it is shown that the period of ionium 
is at least 35,000 years ; and if the autunite examined were 
of very recent formation, this result is to be expected. The 
extremely recent formation of autunite is indirectly borne 
out by the work of Marckwald and Keetman (Chem. Ber. 
1908, xli. p. 49), who could not find any lead in a crystal 
of autunite, though -01 per cent. could have been detected. 
Recently J. A. Gray (Phil. Mag. 1909, xviii. pp. 816 and 
937) has estimated the amount of lead in autunite spectro- 
scopically to be only of the order of :005 per cent. Assuming 


Ratio between Uranium and Radium in Minerals. 349 


a direct change of uranium into lead, this quantity should be 
formed in a million years". The results of Mlle. Gleditsch 
and ourselves with autunites therefore can be taken as con- 
firmatory evidence of the existence of at least one inter- 
mediate body of very long period between uranium and 
radium. Incidentally they indicate that Portuguese autunite 
is considerably more recent than the French mineral, and 
indeed it would appear not improbable that the Portuguese 
mineral has been laid down within a period not very many 
times longer than that covered by historical records. 

In the present state of the subject the possibility has to be 
taken into account that the two «-particles, known from the 
work of Boltwood (Am. J. Sci. 1908, xxv. p. 269), to be 
expelled from uranium may be due to two successive changes. 
The similar low range of these two a-particles is an argu- 
ment, according to Rutherford’s rule (Phil. Mag. 1907, xii. 
p. 110), that the second change cannot be a very rapid one ; 
and we are therefore faced with the possibility that uranium 
may be a mixture of two elements of atomic weights 238°5 
and 234°5, which, like ionium, thorium, and radio-thorium, 
are chemically so alike that they cannot be separated. Now, 
if the element of atomic weight 234°5 had a period at all 
comparable to uranium, the ratio of uranium to radium must 
vary with the age of the mineral, both for very recent and 
for very old minerals, as Mlle. Gleditsch’s results indicate. 
In this case the estimate of the minimum period of ionium 
given in the last paper, which depends on there being only 
one long-lived intermediate body, would not necessarily be 
fallacious, for by hypothesis uranium and its first product 
are so alike chemically that they cannot be separated. 

To obtain evidence on this point the specific a-activities 
of specimens of uranium oxide separated from the three 
minerals were compared. On the view suggested, the specific 
a-activity of the uranium from autunite should be lower than 
that from pitchblende, and the latter should be lower than that 
from thorianite. Within 5 per cent. the specific a-activities 
of all three substances proved to be the same. The small 
differences can be well explained by experimental error and 
by the fact that in some of the preparations the @-radiation 
has not yet reached equilibrium. ‘This, therefore, is fairly 
conclusive evidence that the variations in the uranium- 
radium ratio cannot be due to two successive slow «-ray 
changes in uranium itself. 

Physioa Chemistry Laboratory, 

University of Glascow. 


* The correction to 10,000 years on p. 937 of the same volume must 
be supplemented by a second correction bringing back the value to that 
first stated. 


ste 


BY BOO iad] 


XXXII. On the Statistical Theory of Radiation. 
By Prof. Sir J. Larmor, /.A.S.* 


N the Philosophical Magazine for July (p. 122) Prof. 
H. A. Wilson, in a valuable review of my recent paper 
on the statistical theory of natural radiation f, concludes that 
its procedure does not really evade the main difficulty, that 
an atomic constitution of energy must be implied in such 
investigations. One of the positions advanced in the paper 
was that the magnitude of the element of energy needed tor 
the statistics might be chosen at will, provided the size of the 
elementary cell was chosen in a fixed proportion to it. 
Though such a theory has, and must have on the most 
favourable view, imperfect and provisional features, it does 
not appear to me that Prof. Wilson has established this 
formidable addition to their number, and for the following 
reason. 

Using his notation, the heads of the argument there set 
out, perhaps too briefly }, were as follows. If 8 is entropy 
and W is the number of ways in which the system can be 
arranged in the actual state, then 


S=klog W 
=F log Wa Woot a 


where W,, W,... are the numbers for the parts of the system. 
If the first part contains n, elements of energy each of amount 
€,, contained in N, cells, and similarly for the others, then 
the total energy is 

H=en, + eongt ... €N;. 


The natural state of an isolated system is the one that makes 
S maximum subject to H remaining constant. This requires 
1. oops is 
eae ar oe ee 

€. ON, € ONg 

where €,6n,;=6H,,.... Thus 9 is a quantity the same for 
all the parts of a system which is in equilibrium of exchanges 
of energy: in fact if absolute temperature T is defined by 
the Clausius formula dS=dE/T, then 3 is T—1. Also the 
working out of the actual value of W, leads to 

os N, 

Sa ah loe (1+ 5") 

on : Wy 
* Communicated by the Author. 
+ Roy. Soc. Proc. 1909, vol. lxxxiil. A. pp. 82-95. 


{ Errata should be noted: on p. 92 $ should be T; on p. 93 the factor 
87/A* should be A*/87, 


On the Statistical Theory of Radiation. 551 


Thus, finally, for the distribution of energy among the parts 
of the system we have the formula (Planck’s) 


Nie 
K,=76= Joho 
The argument of Prof. Wilson is that E,(=n,e,) as thus 
determined cannot be independent of the size of the energy- 
element ¢€,, because e, is the only variable that enters except 
N,, which measures the extent of the system, so that any 
change of e, must change the value of E,, even though ¢,N, 
is kept constant: for example, if e, is taken very small, the 
formula becomes 


B,=Ne“T, 


which represents the law of equipartition. But this un- 
welcome conclusion is evaded simply by recognizing that the 
value of & must be some function of the size of the energy- 
element which is taken as the basis of the statistics ; 1t would 
indeed be strange if it were otherwise. If ke, remains finite 
us e, diminishes, the equipartition is not attained unless T is 
very great. We shall find that it is ke, that is to be taken 
as constant when «,, the statistical element for any given type 
of energy, is changed. 

The two independent constants in the formula are in fact 
N,e, and ke, Their ratio N,k~} is equal to the gas-constant. 
That universal quantity, and N,e, (say «) which is the ratio 
of the energy-element to the extent of a cell, are what affect 
the distribution and are thus of pre-determined values ; but 
there seems to be nothing that demands a definite magnitude 
of the energy-element itself. 

On the Boltzmann form of the theory of probability of 
distributions of energy among the molecules of gases, k 
turned out indeed to be the gas constant. On the present 
form of theory, which involves distribution of elements of 
disturbance with their appropriate energies in the containing 
system as mapped out into cells, instead of mere collocation 
of elements with regard to one another, this conclusion need 
not hold. We may probe this point further. It is known 
as a fact that, under ideal conditions, equable partition is 
very nearly attained as regards the translatory and rotatory 
parts of the energy of the molecules of a gas. This requires 
that, if ¢, is the value of e corresponding to each of the 
translatory or rotatory types of freedom, it must prove to be 
so small compared with «, €,,... that the exponent fe,/T is 


352 On the Statistical Theory of Radiation. 


also small; for that is needed in order to lead to this law of 
approximately equable partition, in the form 


i ae — NO ee 


In this special result the value of the element of energy 
e, has become eliminated. Also N,k—! must be the gas con- 
stant R ; and since N,e, must be «, another universal constant, 
we have ke,=a/R. Hence in this case of simple gas-theory 
the value of & should be inversely as the scale of magnitude 
of the elements of energy chosen ; and the size of a standard 
cell should be directly as that element. And this result 
must be universal. 

Thus the conclusion is, briefly, that to render the entropy 
independent of the scale of minuteness of sub-division of the 
statistics, as is natural, we have only to define it as klog W, 
where the value of & (if we decide to retain it in the formulas) 
must vary directly as this amount of sub-division, or inversely 
as the scale of sizes of the elements of energy that are 
employed in the analysis. But, on the other hand, if & had 
the same value whatever be the scale of the statistics that is 
adopted, conclusions such as those of Prof. Wilson regarding 
the magnitude of the ultimate element of energy would 
necessarily follow. 

To connect formally the values of e, thus demanded by 
experimental knowledge for gas-theory, with those that 
obtain for the types of radiant energy, would involve a rather 
long argument. But the present type of theory works out 
for the domain of radiation as above, and it is readily seen 
that it works out for the domain of gas theory on the ordinary 
lines as indicated in the paper referred to; while a bridge 
can be constructed between the two, as there suggested, by 
noting that both for translatory and rotatory motions in gas- 
theory and for radiation of long wave-length, the principle 
of equipartition is practically effective, so that we may take 
advantage of Prof. Lorentz’s train of ideas connecting these 
equipartitions by a calculation of the amount of the natural 
radiation from a thin metallic plate, considered as arising 
from the collisions of the moving free electrons that are 
required by its electric conductivity. 

The existence of another universal physical constant (a), 
in addition to that of gas-theory, has been postulated without 
any explanation as yet. But its existence is independent of 
these statistical theories ; and it thus seems to have come to 
stay in some form or other. In fact it was early pointed out 
by Wien and by Thiesen that the value A»T’, where A» is the 


ort 


EE an ee GR ee a ee ee 
= 
A 


Amount of Thorium in Sedimentary Rocks. 393 


wave-length of maximum radiation at temperature T, and 
which is by Wien’s displacement-law a universal constant, 
suffices and is required, in conjunction with the other re- 
cognized universal constants of nature, to establish an absolute 
system of fundamental units of mass, length, and time ; its 
dimensions are therefore not expressible in terms of those of 
other universal constants, and it must have an independent 
existence of its own. 


Cambridge, July 4, 1910. 


XXXII. The Amount of Thorium in Sedimentary Rocks. 
II. Arenaceous and Argillaceous Rocks. By J. Jory, 
BPO 


1” this paper the results of thorium measurements applied 
to detrital sedimentary rocks are given. The method 
used has been described in previous papers (Phil. Mag. May 
und July 1909). The rock is ground to a fine powder and 
passed through a sieve of 100 mesh to the inch. It is then 
inixed with from 24 to 34 times its weight of mixed car- 
bonates (thorium-free) and fused in a closed platinum 
crucible till effervescence ceases. ‘The melt is thrown while 
fluid into a platinum dish, and what remains in the crucible 
chilled and broken out. ‘The fragments are then ground 
to a coarse powder in a mortar and leached in hot water 
over the water-bath. After standing all night the cold 
supernatant liquid is removed by decantation. The residue 
is ground to a paste in the mortar ; about 100 c.c. of water 
added, and finally 80 to 100 ¢.c. of strong HCl (thorium-free) 
rapidly stirred in. ‘The final solution is seldom quite limpid. 
I have not found, however, that the presence of a small 
amount of precipitate interferes with the liberation of the 
emanation. Known quantities of a thorite solution added to 
such rock solutions, or mixed with the rock-pow der before its 
decomposition in the crucible, produce the seme effect upon 
the electroscope, sensibly, as do limpid aqueous solutions 
containing the same quantity of thorium. 

The alkaline solution, which is poured off the insoluble 
part of the melt, contains very little thorium; in most cases 
none that can be detected with certainty under the con- 
ditions of the experiment, whether the solution is acidified 
or not. The examination of the alkaline solutions has, 
therefore, not been carried out in every case. The investi- 
gation is tedious and generally indecisive, many hours of 


* Communicated by the Author, 


354 Prof. J. Joly on the Amount of 


observation being required (the electroscope being observed 
when the solution is alternately in ebullition and at rest) to 
detect with certainty a change amounting to a small fraction 
of a scale-division per hour. It may, I believe, be accepted 
as certain that what error may arise from confining the 
examination to the acid solution is very small, not more than 
a few per cent. 

Table I. contains only arenaceous and conglomeratic rocks; 
mainly sandstones of various characters and ages. ‘The 
ereensandstone is, of course, a rock differing from the 
others in mode of origin. The sandstones are for the greater 
part constituted of residual quartz or of quartz and telspar, 
derived from older rocks. As the quartz is, itself, probably of 
very low thorium content, itis not surprising that these recks 
are generally poor in thorium when contrasted with many 
igneous rocks. They are, however, much more radioactive 
than the calcareous rocks, in which, in most cases, the 
thorium emanation cannot be detected even when consi- 
derably larger quantities of rock are used in the experiments 


(Phil. Mag. July 1910). 


TasueE I. THORIUM, 
grm.X10—5 per gram. 
Greensandstone. Werl, Westphalia, Cretaceous. (15) ........:....cseeeeee 0:20 
Sandstone. Obernkirchen, Teutoburger Wald. Wealden. (15)............ 0:30 
43 Viotho, Westphalia. Keuper. (10)!..0.......-022se.sea ieee 114 
5 5 % fs (U7) ood ioe cnee gece 1:02 
Es (red). Heidelberg, Baden. Bunter. (20) ..../...-inesememe 0°12 
As Remiremont, Vosges. i; (20): ooo seas teee eee 0-91 
Westhofen, Westphalia. Carboniferous. (15)...............06 0:74 
By Freienohl, bf (15) 3.05 .neee eee 061 


Quartz Conglomerate. Donebate, Co. Dublin. Old Red Sandstone. (14). 0°33 
Grauwacke-Quartzite. Allrode, Harz. Lower Devonian. (16) 
Quartzite (Taunus). Schlangenbad, Nassau. Lower Devonian. (15)... 034 
Sandstone (red). Loch Torridon, Scotland. Torridonian. (19) ......... 0°27 
Quartzite-schist. Western Spessart. Archean (?). (17)........:sesscseeeees 0°32 


In the above table the weight in grams of material dealt 
with is given in brackets. 

The finer-grained detrital rocks—slates and shales—are, in 
contradistinction to the sandstones, derived from the more 
soluble and friable constituents of the primary rocks: such 
constituents as are reduced by denudative actions most 
readily to small dimensions. They are on this account pre- 
cipitated furthest from the land, and represent materials 


Thorium in Sedimentary Rocks. 395 


from which the more resistant grains have been sorted by 
gravity. As the latter are generally quartz or felspar, and 
hence substances which in most cases are poor in radioactive 
constituents, it is to be expected that the argillaceous group 
of detrital rocks would reveal a more considerable richness 
in thorium than the arenaceous. ‘Table Il. shows that this 
is, indeed, the case. Comparison with Table J. shows that 
only two of the sandstones, Nos. 3 and 5, have quantities of 
thorium equal to those generally prevailing in the argillites. 
A few surface materials of recent date are included in the 
table. These possess the same degree of richness in thorium. 


TABLE IT. THORIUM. 
grm. X 10—5 per gram. 

1. Brick-Earth. Rosslare, Co. Wexford. Recent. (18)...........:sscceeeeeees 113 
Pumpelay... Priesdorf, Bonn. Recembid (lo aisteroscces-csscacesel-sad.dece<cerowes 0-91 
Beeetoers. Heidelberg. Pleistocene, (C16) ii cqec vc---c4sceraeoeeenesee-sonceerues 1:04 
4. Bundnerschiefer (folded). Piz Ot, Ober Engadin. Jurassic. (15) ...... 091 
©. Schiste Lustré. Simplon Tunnel. Jurassic, (15) ..........00--ceseeeceeeees 1:04 
6. Red Marl. Ballymurphy, Co. Antrim. Keuper. (12) ................000+- 0-14 
7. Roofing Slate. Wissenbach, Nassau. Upper Devonian. (19) ........... 112 
8. Pe = Caub, on the Rhine. wt _ CG) cue aaeee 1°40 
adate.)) Valentia, Co: Kerry. Devoniane) (1B) icnciies causes oc oeddnenesee odes 1:30 
10. » (dark Killas). Cornwall. _,, (UES he Sd eo eee 1:16 
11. Grauwacke. Wipperfiirth. Rhen-Prussia. Middle Devonian. (15)... 2°40 
12. Clay-Slate. Magdesprung, Harz. Lower Devonian. (15).................. 0:87 
fee, ee Wierlbore. Nassan:'Devomtants (QU tasetecscccc +0 ccecnexeds LG 
14. Slate (green). Kingscourt, Co. Cavan. Silurian. (15) .................s00. 1°30 
15. Phyllite (green). lLossnitz, Saxony. Lower Silurian (?). (15) ............ 1-94 
16. Shale (black). Moffat, Scotland. Silurian & Ordovician. (10) ......... 1:00 
ieehootineg Slate. Penrhyn, N. Wales. Cambrian. (16) 22.........0000- 20.06 0:96 
18. Slate (Oldhamia). Bray Head, Co. Wicklow. Cambrian (?). (15) ...... 0°82 
Meare. ca. 114 


The highest result obtained, the Grauwacke No. 11, was 
checked by a second examination of the preparation, when 
the first result was almost exactly confirmed. ‘The lowest, 
the Keuper Marl No. 6, refers to a material deposited under 
continental conditions, probably in inland waters, and is 
therefore of somewhat different character to the others. It 
contains very little calcium carbonate. Excluding this 
material, the general mean for the argillaceous group rises 
to 1:°20x10-° gram thorium per gram. It will be noticed 
that there is a remarkable sameness in the foregoing results : 
ffteen rocks vary between the limits 0°82 and 1°40. 

The results which I have already published (Phil. Mag. 
July 1909), when dealing with the St. Gothard rocks, are 


356 Amount of Thorium in Sedimentary Rocks. 


reproduced below. These rocks are regarded as for the 
greater part of sedimentary origin, although highly altered, 
and of Mesozoic age. Some are calcareous, some quartzose. 
It is, therefore, difficult to classify them with the materials 
grouped in the foregoing tables. The general means are, for 
the same reason, somewhat misleading. After subtracting 
from thcse of the Usernmulde two rocks of calcareous 
character, the mean for the Usernmulde (9 rocks) is 
1:10x10-°. The mean for the Tessinmulde exclusive of 
the dolomite (7 rocks) 0°53x10-*. The first is in close 
agreement with that of Table II. : the second is too low ; but 
the origin of some of the more basic rocks in the Tessinmulde 
is sufficiently doubtful to deprive the result of much of its 
weight. 


Taste III. 
Usernmulde. Tuonrium. 
gram X 10—9 per gram, 
eeouisern eneiss. (LUT ic... Seven eaters ou eae 14 
Baeamartz-sehist, (9°6).....c.s.s-omerentastes, soaeean es 
Seaoeeack lustrous plate. (SQ) .\ceviwenen ss aoe 0:2 
Someetey CIPOliN, (C927) a. vitssscumhoromuouantewaaceas 0-4 
Seppauartzitie cipolim, “CUIGO) ; tessa. eye naeea yee O-4 
O-jepack lustrous, slate; (10°66), ys .vseonk ts ween 10 
foe wisent eneiss, (O71) cect etius us wate scien oc camen 13 
PemeCericibe-SChish. | (SO): rx aca saomues cemmera nascent 7 
y. Black lustrous schist (7784 jes seve cnnate ccs oaen 2'4 
Ae wuevaartz—noiea, (Sik G) eins -cuas. imecceemesass se) <0°3 
MlsyVUisern mica-gneiss. OOS). .cc tye seeks 05 
Tessinmulde. 
2.) ALornblende-sehist. | (8/64) ssn. 21 sceeagese< sess nes <=0°3 
13. Calcareous mica-schist. (9 02) .............0000. 0°5 
14. Hornblende ,, _,, Wed Spee ta eee es 0°6 
15. Amphibole garnet mica-schist. (8°38)......... 1:0 
16. Quartz-schist Oi er ecateemes thi soe. <=0°3 
17. Amphibole mica-schist. (9°17) .2...5.2.......00 0-5 
18. Quartz x bh) SBD A etre Faces cathe 0-5 
19. Dolomite. (St6G)) Sacdedon as Sia zaet 0-4 


If we assume that the results on sedimentary rocks, recorded 
in this and the previous paper on the subject, may be accepted 
as approximately representative, we find that whereas the 
calcareous rocks show a small, almost negligible, quantity 
of thorium, the detrital sediments contain easily measured 
amounts of thorium in almost every case; the argillaceous 
group having almost double the amount contained in the 


Magnetic Balance of MM. P. Curie and C. Cheneveau. 257 


arenaceous group. The former may be taken as approxi- 
mating to 1:3x10-°, making allowance for some small 
amount in the alkaline solutions; the latter to 0°6 x 10-° gram 
per gram. 

Accepting the estimate cited by F. W. Clarke (‘A Preli- 
minary Study of Chemical Denudation,” Smithsonian Miscel- 
laneous Collections, vol. lvi. No. 5, June 1910) that the 
calcareous rocks compose 5 per cent., the arenaceous 15 per 
cent., and the argillaceous 80 per cent. of the sedimentaries, 
my results on thorium measurements (assuming 0°06 x 10~° 
to represent the mean for the calcareous rocks) give for the 
sedimentary rocks generally a thorium vontent of 1:16 x 107° 
gram per gram. 


July 11th, 1910. 


XXXIV. The Magnetic Balance of MM. P. Curie and 
C. Cheneveau. By C. CHENEVEAU, with an Appendix 


by A. C. JOLLEY *. —  f) fi 
FEXHIS apparatus is intended for the measurement of the 
e coefficient of specific magnetization, or the suscepti- 
bility or permeability of feebly paramagnetic or diamagnetic 


bo:lies Tf. 


Principle and Theory of the Apparatus. 


The body whose magnetic properties are to be determined 
is suspended from one end of the arm of a torsion balance. 
By means of this balance the force is measured, which 
is experienced by the body when placed in a non-uniform 
magnetic field, produced by a permanent magnet whose lines 
of force cross the space occupied by the body. The method 
of calculating this force will first be briefly indicated. 

Suppose that the body is placed at a point O in a field of 
direction Oy and of intensity H,. The force f which tends 
to move the body will be normal to the direction of the field, 


* Communicated by the Physical Society: read April 22, 1910. 
+ The coefficient of specific magnetization K. is the ratio of the 


: : Bate. M : 
intensity of magnetization = =) (Where M is the magnetic moment 


and m the mass of the body) to the magnetizing field. The magnetic 
susceptibility k=KA, where A is the density of the body, and the 
permeability is obtained trom the susceptibility by the relation 
p=1-+4ik. 


398 M. C. Cheneveau on the Magnetic Balance 
2. e. in the direction Ow (fig. 1). If & is the intensity of 


specific magnetization, the value of the force is * 


fame +. ttbass date =: ae 


oe representing the space variation of the field. 
Fig. 1. 
Bcd Hy y 


j 


Me 


As we are only concerned with feebly magnetic bodies, 
the demagnetizing force arising from the magnetization of 
the body is negligible, and we may assume that the intensity 
of magnetization is proportional to the field. If we denote 
the constant ratio between the intensity and field, or coefficient 
of specific magnetization, by K, we have 


JaKE ie i 


consequently combining equations (1) and (2) 


oH, 
f=KmH, a .. 7. ee 

Let us first suppose that the magnet producing the magnetic 
field is at a considerable distance from the body. Then 
H,=0, and by (3) the force is zero. 

The body being always situated at O, let the magnet be 
brought up to the position I (fig. 2). If the force fis one 
of attraction the body is of course paramagnetic, if of 
repulsion, diamagnetic f. 


Ww 
* We have f= °- 4 W=MHE, and M=Jm. 


oH 
Hence 7 oe, 
+ Ifthe sense Ov is taken as positive and we employ the true formula 
for the force, f=— od the negative sign for the torce indicates 
ros 


attraction and the positive sign repulsion. 


ee ay: 


Jaco, = eels 


of MM. P. Curie and C. Cheneveau. 309 


Suppose first that the body is paramagnetic. 

Whea the magnet is situated so that the body O coincides 

: eat ia ee 
with the centre G, the force will again be zero, as oo is 
zero at the centre (position II, fig. 2). A 


Fig, 2. 


Hence the force f is zero when x=x and x=0 and it 
passes through a maximum in the interval, which occurs at 
a certain point O’ where the product of the field H, by its 


e oH . e 
gradient re is a maximum. 


If the magnet is continually turned in the same sense of 
rotation, so that its centre G passes through the point O and 
arrives at G’ (position III, fig. 2), we observe that the force, 
which was nil at O, reaches a minimum or negative maximum 
at a point O” symmetrical with O’, relatively to O, and 
becomes zero again after the magnet is withdrawn far from 
the body. 

Hence the curve in fig. 3 shows the variation of the force 
jf with the displacement 2 The body is first displaced in 
the direction of the arrow marked 1 from O to O’, where it 
stops and moves back with the magnet along the arrow 


360 M. C. Cheneveau on the Magnetic Balance 


marked 2 to O'’, where it again stops and afterwards returns 
to O, as shown by the arrow 3. 


It is easy to explain in a similar manner the motions of a 
diamagnetic body which will of course be the reverse of the 
above, as the force is opposite in sign. Fig. 4 shows the 
curve in this case. 

Fig. 4, 


5 


Sn 
DiREctioN oF Motion oF MAGNET 


At every instant the action of the force is balanced by 
the torsion of the balance wire. If ¢ is the moment of 
torsion per unit angle, and « the maximum deviation of the 
arm at O or O/ (fig. 2), the equilibrium condition is 


Simca, 67s. . | ae 


if Z is the length of the balance arm. 
If the deviation is measured by lamp and scale, 


D 
eR RE 


D being the deflexion of the spot upon the scale, and L the 
scale distance. 


— 
en 
‘ 


of MM. P. Curie and C. Chenerear. d61L 


Hence From equations (3), (4). and (3) 
q : 


KmH, les EAE ION WOES aN Aa 
from which <D | 
| Ges pre ak (7) 
2mlLH, rain 


This formula (7) thus permits the absolute value of the 
coefficient K to be determined for a body if the other 
quantities are accurately determined. By an analogous 
method P. Curie * has determined the absolute value of the 
coefficient of specific magnetization of water. iahie 

But the apparatus is especially adapted for relative 
measurements, and in this case it becomes extremely simple 
and practical. 

If we have a body of unknown coefficient K and of mass 
m, we have from equation (6) 
ie a! hes 


KmH, ~ l= oT,’ 


For a body of known coefficient K’ and mass m' occupying 
the same volume 


K’m'B, l= : (UE aeseens 


Hence by division 


| ou) Se 8 
Keo he ae (10) 


Equation (10) thus enables K to be determined. 

It is evident that it is much more convenient to measure 
the displacements D and D’ of the spot corresponding to the 
angular deviations 22 and 2a’ between the two positions 
O' and O” of the body (fig. 2), for, without altering equation 
(10), the displacements of zero are eliminated and _ the 
accuracy of measurement doubled. 

If the body is placed in a tube of glass or other substance, 
this tube being subjected like the body under test to the 
magnetic forces, an experiment must be made with the tube 
alone, aud the effect due to the tube represented by a 
displacement of the spot D” added or subtracted. 


* P. Curie, Annales de Chimie et de Physique, 7° Série, t. v. p. 344 
(1895). | 
Phil. Mag. S. 6. Vol. 20. No. 116. Aug: 1910. . 2B 


362 M. C. Cheneveau on the Magnetic Balance 


We thus have 
Kem DD" 
K! e an’ — p =D" ee e © e ° (11) 
Whence Ki sy D +D" m! 12 
K' — D'+D" e m e e ° ° . e ( ) 


This formula is not corrected for the effect due to the 
magnetization of the air ; the exact formula may be obtained 
as follows: 

Let «' and x” be the susceptibilities of the substance taken 
as standard, assumed to be paramagnetic, and of the air 
respectively. Let A’ be the density of the comparison body, 
and A a constant of the apparatus. In reality the exact 
expression for the force when one deals with the standard 


body is 


! /I 
Mm K : 
{fp /t SmA Gifs oe Bd Vener 9 
fi=(K -k dal A= (K “ym A, ot aye 

since r 
K 
AM 

When a measurement is made with a paramagnetic body 
of susceptibility « and density A the true value of the force 
in this case is 


i 


i 1/ 
f=(K—-k’’) RAS (K-K) mA. er ie 
Dividing (14) by (13) we have 


tf 


ae 
fit 7D Bm 
are Ke ei 
we 
or He 
pot Boa 
fa = Tog: * + 
Wie) K' K 
Si 


! 
If we put jae. = ; this is the approximate ratio of the 


coefficients of specific magnetization K and K’' previously 
determined by the aid of formula (12) *. 


* We have in fact, from what has been proved above, 
fl=KmA=B(D+D"), 
fl=KmA=BW ED"); 
A and B as well as / being constants ; whence 
i he DED” fit AK Dasa 
7 ha DED?! fa KT Dee ae 


of MM. P. Curie and C. Chenevear. 36: 


From (15) el! 
ue K— A 
oe 


> K y K 
K— x = rh Pal 3 
. K Kl! a ee 
ep are 

nm RA KA 


- ay 
= an E i KART al bs Jee GG) 
in which eA pep mi! 
D’+ DD" m~ 
When the comparison body is water 
K exactly ==079 x 10-° (P. Carie). 


As for air the susceptibility «’”’=0°0322 x 10-*. 
It r is positive we have for a paramagnetic body 


K fi 1 
pent [1+ 0-041 (+1), 


and for a diamagnetic body 


K 1 


Hence the apparatus lends itself very readily to relative 
determinations, and if we take water as the standard bedy » 
(K’=-—0°79x10-*) it enables us to obtain an absolute 
value of the coefficient of specific magnetization of a body. 

But the comparison substance may be a liquid or solid 
other than water, and such that its coefficient of magnet- 
ization is of the same order of magnitude as that to be 
determined. A choice can be made from the tables of 
magnetic constants. It is convenient if the masses of the 
bodies compared correspond to the same volume; on this 
account the tube containing the substance is always filled to 
a given mark. When the substances tested are very strongly 
magnetic, a smaller length may be employed, but in that 
case it is absolutely necessary that the comparison substance 
should have the same length. 

The apparatus could be made still more sensitive with the 
aid of an electromagnet. On the other hand, with a torsion 
wire of larger diameter, the magnetic properties of iron or 


2B2 


364 M. C. Cheneveau on the Magnetic Balance 


other ferro-magnetic metals and alloys could be determined. 
It would doubtless suffice to use an extremely fine wire of 
the metal, and to employ the method of experiment and the 
corrections already proposed by P. Curie in a work on the 
magnetic properties of bodies (loc. cit. p. 5, note 1). 


= SS == ——— as 
———— SS SSS S| 


ll 


(i If Filitnces 
Mil as 
| 


) 


N 


in SIP Sane 


ue hlUGt~— OR 


Description of the Instrument, Magnetically Damped Form. 


The arm of the torsion-balance is formed by a rod TT 
(fig. 5), which carries from one of its two extremities a 
hook ¢, from which the glass tube ¢, which contains the body, 


of MM. P. Curie and C. Cheneveau. 365 


ean be suspended. To this end the glass tube is closed by a 
cork carrying a metallic ring which hangs on the hook ec. 
The tube ¢ is therefore in a vertical position. On the other 
end of the arm TT is fixed a copper sector, S’, which moves 
between the poles of a fixed damping magnet A. To vary 
the sensitiveness the torsion-wire can be changed, and the 
damping varied by the position of the copper sector in the 
field. 

On the horizontal arm TT is fixed a vertical eoppe strip 
which carries a hook to which the torsion-wire f (5 mm. 
platinum) 1 is attached. Below, this strip is turned ab right 
angles as shown by the portion DE paraliel to TT. Balance 
is obtained partly by the aid of a small copper cylinder B, 
sliding on the rod TT, when the tube is empty, and partly by 
copper or aluminium riders, pp, to compensate for the weight 
of the solid or liquid material which fills the tube to the level 
marked. 

The displacement of the magnet N.S. is obtained hy a 
movement of rotation around an axis O; this movement is 
smoother than the sliding motion of our earlier model. The 
movements of the torsion-balance are followed by the observer 
upon a divided scale C. (fig. 6). For this purpose a 2-metre 


Fig. 6. 
C 


TRAUVTTUYANUUCUOANOSCOTRSOUHANV UAE HAUTE cy | 


| Pond daha! Pudi epindadahaday Lrllatlaituh (il eiaidlabdielsi tate dil | 
TEROOUYUYUAOUOUVANUTSEAYORAURSTTOEUAOUOTTVROEEUO THOTT PE Se NUN an iii el) 


2 es) 
i Ny 


9). 


HIE: 


7).__pp, 
yt 
=== Se 


—— oo 


P = 


— jae 


aa li 


radius concave mirror m is pe oneal to the balance-arm, and 
the ima ge of a straight filament is employed. The displacement 
of the. magnet is effected by.a controller M (fig. 6). fixed in 


366 Mr. A. ©. Jolley on the Magnetic Balance 


front of the scale, which consists of a horizontai rod PQ 
turning about a vertical axis in a heavy base, and furnished 
with a large milled head G, which can be turned by the 
observer. This rod is provided with two clamps P and Q, 
which are attached by two cords to two rings R and V on a 
similar rod attached to the axis of the magnet NS. If the 
cords are initially arranged so that the rods PQ and RV are 
parallel, the position of the magnet is at once determined by 
the direction of the rod PQ, and it can be turned either by the 
milled head G, or preferably by holding the two cords ss like 
reins. 

The advantage of employing a rectilinear filament lamp 
(such as is made by the Pintsch Gluhlampenfabrik in Berlin) 
is considerable in practice, as the spot is always visible on 
the scale even if the torsion-arm is not perfectly balanced 
horizontally. 


Appendin. 


By the courtesy of M. Cheneveau and the Cambridge 
Scientific Instrument Co., we have been able to set up and 
test the performance of one of these balances in the labora- 
tories of the Northampton Institute. 

The instrument was of the magnetically damped type 
described above, and was used with a lamp and scale at some 
two metres distant from the mirror. 

No attempt was made to obtain absolute values of the 
coefficients of specific magnetization; but relative deter- 
minations only were made, using as a standard substance 
distilled water, and assuming for it the value 


K' = ere & 10-*, 


as found by M. Curie and given above. 

The results obtained are set out in the accompanying table, 
and show some interesting figures. 

The first portion of the table contains figures for a few 
materials selected quite at random in order to test the 
behaviour of the balance. Of these the first five are 
chemically pure liquids, and all exhibit diamagnetic qualities. 

The four samples of aluminium are all magnetic, and we 
have not been able to get a sample which is less magnetic 
than the first of these. The eleven samples of brass indicate 
the range over which the magnetic gualities vary, while some 
of the samples examined, but not here tabulated, have been 
too magnetic to be used in the apparatus without changing 
the suspension. 
~ It is interesting to note that during some inductance tests 


————|——_+_ 


Be Sae bcs eae (| 


Denke ffect of Volume on 
4 Value of K. | 


[Zo face p. 366, 


Effect of Length on 
Value of K. 


87 NicxeE tin, I. 


87 


1885 
1187 
ees, 2 8-9989 
Pee avin ok 89789 
9-0592 
8°9393 
Ae, eae . 192 
oie oe oe 8-9 


Bee Ge ee ee a 


| 204x 10-8 


1-99 
2°16 
2°25 
2:03 
2°11 
2-13 
2°14 
2°18 
2°18 
2°41 


SAAR Ft deci 86 
belenhutte .. 86 


BS Oe 8 eet ies) me 


8-6 ExtTRA PRIMA. 


Pe Ball bene 
1:52 
16 

1:52 
1:66 
1:29 
1:38 
1-29 
1:28 
1:19 
1:23 
1:13 
1:07 
1:09 


CopPER. 
Length. K, 
5 cm. |0-404x10~° 
4:5 0-446 
4:0 0°516 
3°50 0:642 
30 0-828 
2°5 0:71 
2:0 —0°895 
15 1-093 
1-0 (1-46 


05 2:37 


EFFeEect oF ACID 
TREATMENT. 


CopPER. 


Mass, m. Ke 


2°545 gr. 


1:16x 107° 


After Treatment with Hot 
Cone. HCl :— 


2°542 | 0°765 
| 
Further Treatment with 
HC] :-— 


2540 


Cupnnvnau & JOLLBY. 


. Density, 
SUBSTANCE. ne 
Witenes 10 
INGO ooo cones 079 
Amyl Alcohol ..| 0810 
Benzene......-- 0:85 
Luxor Oil ...... 0:8 
Methyl Hther 0725 
Aluminium 2:67 
” 
” 
” 
IBGE sososoouce 81 
hy soot ord coy 
Saath Coreen ons 
Sif mace oro cin 
ee RO ONS 
Pe ance bape 
AMT A ieisstes a 
o) coon youd® . 
(Ghoysyar con soon oo 
nit LigeEouee 
poems 
» 
= cReSOADDD 
Phy DSC ORL 
9500 ke el 
teh CROC O G 
OOO OG oe 
MW Sooustresooes 73 
Vwi Goaret cone 6:9 
yh) CO ODEO OO 
» (Pure) 


0:707 x 10-* 

0-455 

1:06 

0:68 

0:54 

0:36 

| 0:83 
0:537 

—0:097 
0151 x10—-° 
0043 x 10—* 
0:212 
0-224 


Coefficient 
of 
Specific SUBSTANCE. 
Magnetization, 
K. 
20/790 wail pNickeliny leone 
—078 Nickelin, Dic... 
—()'929 Woden, ) dennnescoann 
—0'962 Wakelin, U3) doses naoaesc 
—0:594 JOpsxtoye, LEWIN, | Son coneo ns 
—(0:785 PVM oodobnoorooane 
mi i padleeoogen caps 
1:19x10-° Fens Uti acon anes 
179 Constantan Vogel ...... 
2-03 ca: | eee 
2:99) ci | See 
—0:02610—* | | Constantan, 12.......... 
3°95 ILS So Sepak esas b odo 
0:3 German Silver, 2A ...... 
0:09 New, Metall }innractaa: 
0-088 | | | eee 
2:02 Breaker ciel see een tver 
194 Manganin, 14) (yieeas se 
0:282 Manganin Isabelenhutte . . 
3:09 IVAN Gadsooocacuane 
4:05 SHPISWOP os odloagocann 
2°45 


Specific 
Resistance, | Temperature 
Density, IK. Microhins Coetlicient 
A. per em. cube Ns 
at 0° C. at 17° C. 
p. 
9:09 488x10-°} 43-13 0:00245 
877 375 40°62 0-021 
879 Pritt 39°29 0:0187 
8:82 a76, 39:73 0/021 
88 All 29°35 0:028 
8:83 175 33°62 0:0224 
161 
ae 1:68 
8:99 9:67 47-06 —0:00122 
10:25 
Ke 9°72 
8:97 12°37 48:3 0:001437 
8:97 9:23 50:7 —0:0029 
9:06 677 42°89 0:00385 
8:97 6:87 51:10 —0:0088 
5b O77 
8:96 ie 47-4 0 0048 
8:61 28:05 36°62 0:00175 
8:6 28:7 39°14 0:00176 
8:61 47-55 0-024 
8:24 81-024 01148 


to Cu 
Microyolts 
per ° C. 


BYA) 


Thermo E.M.F. 


Effect of Volume on 
| Value of K. 


[Zo face p. 366. 


Hitfect of Length on 
Value of K. 


| Nicke tn, I. CoprEr. 
| 
| = = 
Mass, m. | K. Length. K. 
| eee ee ae ee 
| O98 | 204% 1056 5 cm. | 0404x107 
1985 | 1:99 45 0:446 
1784 | 2:16 40 0516 
1583 2:23 30 0:642 
1385 2:03 3:0 0:828 
1187 211 25 071 
0:989 2:13 2:0 | 0:895 
0789 2:14 15 1-093 
0592 2:18 | 1-46 
0393 2:18 05 2:37 
0192 241 oes eee 
Mrrect or Acip 
‘TREATMENT. 
Exrra Prima. a 2 
Coprrr. 
0:21 Pil s<TO=® : 
0-434 159 Mass, 7. K. 
0635 16 a 
0:836 1:52 2:545 gr. | 1:16 x 1@—-” 
1-033 1:66 SS ae 
1233 129 After Treatment with Hot 
1435 1:38 Cone. HCl :— 
1637 1:29 
1:835 1:28 2542 } 0765 
2033 | 1-19 Ss 
9:93) Wk 
ie ae a ei J eaten’ with 
2:684 1:07 = 
2854 1:09 2540 | O774 
| 


i ee 


of MM. P. Curie and C. Cheneveau. 367 


in progress in the laboratory, circumstances compelled us to 
use brass conductors, and the results obtained were higher 
than those given by calculations based on theory. This could 
only be accounted for by assuming a permeability for the 
brass greater than unity, and although the magnetism in the 
metal was not detectable by a fairly sensitive magnetometer, 
it was readily shown when samples of the conductors were 
tested in the balance. 

Nine samples of copper are tabulated, and one only is 
diamagnetic, again emphasizing the difficulty of obtaining 
really non-magnetic conductors. 

The tin sample was chemically pure and is also magnetic. 
Of the three samples of zinc the first two are ordinary com- 
mercial sheet metal, and the third one is a rod of chemically 
pure metal for use in standard cells. 

The next portion of the table is devoted to an examination 
of the series of resistance alloys, whose electrical properties 
were investigated by the writer some time ago and published 
by Dr. Drysdale in connexion with his paper upon the 
Comparison of Standard Resistances, before the British 
Association at Leicester, 1907. 

The first point which strikes one in connexion with these 
is the fact that every alloy is magnetic, but that those alloys 
containing relatively a large proportion of the magnetic 
metal nickel, 7. e. Nickelines, are among the least so. The 
first sample is very different from the other three, and we 
have reason to believe that the second and fourth samples are 
of the same manufacture although obtained from different 
firms. 

The samples of Platinoid fall into this group with properties 
very comparable with the Nickelines, and the material sold 
under the trade name of Extra Prima probably also belongs 
to this group. 

The Constantans come next in order, being more magnetic 
than the Nickelines, and it is not difficult to see that the 
material catalogued as 1A1A belongs to this group of alloys, 
together with the alloy supplied to us under the title of New 
Metal, while the sample German Silver 24, but for its high 

density and positive temperature coefficient, would also be 
classed among the Constantans. 

Eureka is a resistance material which is often classed as a 
Constantan, but the balance easily disproves this, as it shows 
it to be so magnetic that observations would have had to be 


made with a stiffer suspension than we were using, and this, 


coupled with its positive temperature-coeflicient, rule it out of 
the group. 


368 Mr. A. C. Jolley on the Magnetic Balance 


Superior and Rheotan behave like steels in the apparatus 
and probably contain considerable quantities of iron in their 
composition. 

The magnetic behaviour of Manganin is remarkable, it 
being, with the exception of the last two alloys and Hureka, 
the most magnetic of all the materials examined. This is of 
particular interest in view of its almost universal adoption 
for the construction of accurate resistances, but it is not 
altogether unexpected when we remember that the remark- 
able Heusler alloys have a Copper-Aluminium Manganese 
composition. 

The figures in the rest of the table indicate the variation 
in the value of K, with increase in mass of the specimen and 
increase in length. In the first case the sample, in the form 
of a bare wire 1 mm. in diameter, was cut up into standard 
length (approx. 2°8 cms.), and successive lengths of measured 
mass were introduced into the tube until it would hold no 
more; it will be observed that the value of K tends to di- 
minish as the mass increases. (Curve A, Extra Prima, fig. 7.) 

The apparatus is, of course, very sensitive to the length of 
specimen, and an extreme range of length was taken in a 
wire of 1 mm. diameter, and the results are shown in the 
curve B (copper). 

The difficulty of obtaining non-magnetic copper has already 
been referred to, and every instrument-maker knows how 
difficult it is to wind a really non-magnetic D’Arsonval coil. 
In order to see how far this is dependent upon surface 
conditions a sample of copper was taken and treated with 
concentrated hydrochloric acid and re-tested, and it was 
found that the value of K was reduced to nearly one-half, 
and that further treatment had but little effect. 

From these results it will be seen that there is a consider- 
able field of usefulness for the balance. Firstly, for examining 
the materials to be used in the construction of standard appa- 
ratus where the permeability becomes of first importance. 
Secondly, in grading and examining alloys whose properties 
are materially dependent upon minute traces of alloying 
constituents which are so difficult to estimate by chemical 
means, and which play such an important part in their 
electrical behaviour. Thirdly, it could well be used to 
maintain the standard of purity in the commercial pro- 
duction of such a metal as Aluminium, whose chief im- 
purity is iron, the last traces cf which are so difficult to 
remove. deat : 


3 4 5 


LENGTH IN 


In conclusion the writers’ thanks are due to Mr. A. F. 
Burgess, B.Sc., for his kelp in the experiments and 
calculations. 


XXXV. Typical Cases of Ionization by X- Rays. By CHareEs 
G. Barkua, M.A., D.Se., Projessor of Physics, King’s 
College, London * 


ue a letter to ‘Nature’ (April 15, 1909) and in a short 
preliminary paper on “ Phenomena of X-Ray Trans- 
raission ”? (Proce. Camb. Phil. Soc. May 17, 1909) the writer 
showed that many apparent anomalies of ionization might be 
explained in terms of a few simple laws, and that so far as 
experiments had then gone, the behaviour of one substance 
might be regarded as ty pical of all. Further experiments 
have confirmed these conclusions, but as accurate measure- 
ments of ionization involve the study of absorption, secon dary 
X-radiation and secondary corpuscular radiation, the publi- 
cation of the results in detail cannot yet be undertaken. It 
is sufficient for the present to study the results of experi- 
ments on two substances, carbonic acid gas and ethyl bromide 
vapour, as these show all the marked phenomena hitherto 
observed. 

Secondary X-radiations from Fe, Ni, Cu, Zn, As, Se, Br, 
Sr, Mo, Ag, Sn, and Sb—with special treatment in some 
cases—furnished homogeneous beams of X-rays which varied 
considerably in penetrating power. The most penetrating 
radiation dealt with was seventy times as penetrating as the 
most absorlable. The absorption of these radiations by Al, 
together with the substances which emitted them, is shown 


. . . —AJ . 

in Table I. 2 is defined by the equation [=Iye ™ during 
. . e . xX e J . 

transmission through aluminium, — is tabulated because it 


is a constant for a given radiation and given absorbing 
substance whatever the density of that absor bing substance. 

In studying the results of experiments on ionization it is 
necessary to consider the ionizing effects of :— 


(a) Radiations from the walls of the ionization chamber : 


(1) Scattered X-rays—(one type of secondary X-rays); 

(2) Transformed X-rays—(secondary X-rays character- 
istic of the radiating elements) ; 

- (3) Corpuscular ‘r rays—(secondary rays consisting of 
ejected electrons). 


* Communicated by the Author. The expenses of this research have 


been partially covered by a Government Grant through the Royal 
Society. 


Typical Cases of Ionization by X-Rays. 371 
()) Radiations from the gas or vapour studied : 
(4) Seattered X-rays ; 
(5) Transformed X-rays ; 
(6) Corpuscular rays. 


A few notes on each of these are necessary before 
attempting to interpret the results of experiments :— 

(1) The energy scattered by light-elements has been 
studied by the writer. This radiation being of the same 
type as the primary produces the same relative ionizations 
as the primary. As it is not directed in a parallel beam, 
however, there must be a small error in the correction for 
absorption as applied to the primary beam. The error 
introduced is, however, exceedingly small. 

(2) No characteristic X-radiation of penetrating power 
between the extreme limits of the primary radiations used 
is emitted by Al—the material of the ionization chamber. 
Extremely “soft” radiations, if they produce an appreciable 
iunization, are included in the correction applied tor the 
corpuscular radiation from Al. 

(3) The secondary corpuscular radiation from Al produced 
an appreciable ionization in some cases. Correction has 
been made for this from the results given by Mr. Sadler *. 
The maximum correction was about 20 per cent. of the total 
ionization. 

(4) The energy of the rays scattered by the gas was, 
in these experiments, always a negligible fraction of the 
energy of primary radiation and consequently produced a 
negligible ionization. 

(5) The secondary X-rays known to be emitted by the 
elements in ethyl bromide produced in the ionization 
chamber used not more than about 1 per cent. of the total 
ionization. The ionization produced by exceedingly soft 
secondary X-rays is unknown. It is included in what will 
be called the direct ionization of the X-rays. 

(6) The corpuscular radiations from the elements in ethy] 
bromide will be discussed later. 


The homogeneous beams of X-rays were passed through 
air, O, CO, SH,, SQ, coal-gas, N.O, C,H;Br, CH,I, 
Se,Cl,, SnCl,. After correcting for absorption in the ionized 
gas, and for the effects of secondary corpuscular radiation 
from the walls of the ionization chamber, the results given 
in the following Table were obtained from experiments on 
air, carbonic acid, and ethyl bromide mixed with air. 


* Phil. Mag. March 1910, pp. 637-356. 


“a 


Prof. C. G. Barkla on Typical 


TaBLE I. 

Source of |4Dsorption of Toutata a 
homogeneous ae Toniz. in CO,. Toniz. due to C,H;Br, ie at O,H,Br at 
X-radiation. | in Al (;) - | Loniz. in air Joniz. in air baie: if vcr! 

P ee pressure, 
[Numbers purely relative } 
MG ticle 88°5 1°42 29°5 1257 571 
Ne eM 59-1 1-385 30 | sly 387 
Gah Lac eae 47°7 1:39 30°2 as 315 
LU se ee 39°4 1:36 31-1 53°6 268 
(ee ae Sek Ee 22:5 1376 30°2 31 1485 
SI eer 185 1°35 2058 25 120 
Ba Hike eee 17°4 - 30°9 — 117°5 
ST Leaves teey. 13 14 106°7 18°2 303 
LT ae ae 4:7 1-42 153 6:7 137 
1 ag ae S| 2°5 1:38 1538 3°45 83°5 
SBN cence ciesies 1:57 1-4 175 2°20 60:2 
BSD westtiecie sc 1:2 1-42 166 1-72 44 


* The majority of these absorptions were determined by Barkla and Sadler, 
that for Br radiation by Chapman. 


Column 1 gives the source of the ionizing X-radiation ; 
column 2 the absorbability of each radiation in aluminium ; 
column 3 the ratio of ionizations* in carbonic acid and air 
at the same pressure and temperature ; column 4 the ratio 
of the ionization due to ethyl bromide in air saturated with 
ethyl bromide vapour at 0° C. to the ionization in air at the 
same pressure (11°6 cm. of mercury) and temperature as the 
ethyl bromide. [Possibly this is not equal to the ratio in 
pure ethyl bromide and air at the same pressure. The 
ionizations in vapours when pure and when mixed with gases 
will shortly be dealt with. ] 

An examination of the above results shows that the 
ionization in carbonic acid gas is proportional to the 
ionization in air throughout this wide range of penetrating 
power. Similar results have been obtained with O, SH,, 
SO,, NO, and coal-gas, though in some of these the 
variations from proportionality were greater than in COQ). — 


* The term ionization is a somewhat ambiguous one. Unless other- 
wise stated the relative ionizations in gases mean the relative ionizations 
in films of the gases, so thin that the change of wa of the beani 
of X-rays in transmission is negligible. ‘ 


Cases of Ionization by X-Rays. 373 


A similar relationship is found in the case of C,H;Br 
for radiations not more penetrating than the secondary 


X-radiation characteristic of Br. . ie in Al=17-4. | 


As the radiation is made more penetrating, the relative 
ionization rises rapidly at first and more slowly afterwards. 
It probably approaches a constant though much higher value 
than was obtained with the more absorbable radiations. 

Similar features have been observed in CH3I, SnCl., and 
Seo! ‘l.. | 

These variations in relative ionization are similar to the 
variations in absorption of the ionizing radiation *. For the 
absorption in a given element is approximately proportional 
to the absorption in other elements until the radiation 
becomes of more penetrating type than that characteristic of 
the absorbing element, when it begins to rise rapidly and 
finally approaches a higher proportionality. 

Similar changes take place in the intensity of the secondary 
homogeneous radiation T emitted by an element when subject 
to a primary radiation whose penetrating power is gradually 
increased, except that primary radiations softer than that 
characteristic of the element exposed, excite no secondary 
X-radiation of this particular penetrating power. 

Mr. Sadler { has recently shown that as the penetrating 
power of the primary radiation increases, secondary cor- 
puseular radiation begins to be emitted at the same critical 
point by the particular element subject to the radiation. 
The intensity increases rapidly with an increase in penetrating 
power at first, then much more slowly. 


ionization in CO, . ionization in C,H, Br 
ionization in air "ionization in air 

are shown in fig. 1, in which these ratios are plotted as 
ordinates—(on different scales for convenience)—and the 


absorbability of the radiations e in Al) as abscissee. It 


The variations of 


will be noticed that owing to crowding of very penetrating 
radiations at one side, the curve for C,H;Br turns upward 
again, 

As the ionization in air diminishes with increased pene- 
trating power of the radiation, the variations in ionization 


* Barkla and Sadler, Phil. Mag. May 1909, pp. 739-760. 

+ Barkla and Sadler, Phil. Mag. Oct. 1908, pp. 550-584; Phil. Mag. 
May 1909. i 

{ Sadler, Phil. Mag. March 1910, pp. 337-356. 


374 Prof. C. G. Barkla on Typical 


are more correctly indicated if we assume that the absorption 
of various beams of X-rays in air is proportional to the 


Fig. 1. 

4 200 
& 
g 
S 
S 3 150 
a 
= Go 
x 
S 
& 
= (2 100 
Ss 
S 
S eS a o 
Ss 
wy | 50 
S 
& 
N | : 
Dy 
& 

fe) 60 75 30 


ABSORBREILITY OF X-FA DIATION [4 /N Ac] 
load 


ionization produced in air. With this assumption we get 
the relative values for the ionizations produced by the various 
primary beams of equal intensity as given in columns 5 and 6 
(Table I.). 

The curves exhibiting the relation between the ionization 
in a thin film of gas or vapour and the absorbability of the 
X-radiation are given in fig. 2. The accuracy of course 
depends upon the truth of the assumption made, but of the 
marked features there can be no doubt. These curves 
should be compared with those showing the variation of 
absorption in various elements with variation in penetrating 
power of the X-rays used *. 

The marked deviation from simple proportionality of 
ionization in C,H;Br to ionization in air or to absorbability 
of the primary radiation-is thus due to the presence of Br, 
which has a characteristic radiation within the range of 
penetrating power used in these experiments. 

Similar deviations occur in the case of CH;I at the 
particular penetrating power characteristic of the secondary 
radiation from I, in the case of SnCl, at that characteristic 
of Sn, and in Se,Cl, at that characteristic of Se. Carbon, 


* Barkla and Sadler, Phil. Mag. May 1909, pp. 739-760; Barkla, 
Proc. Camb. Phil. Soc. May 1909, pp. 257-268. 


Oases of lonization by X-Rays. STS 


hydrogen, and chlorine have no characteristic radiations 
within the range of penetrating power used, 


us 
(=) 
oO 


SON/ZATION . 


Mm 
o 
oS 


100 


0 45 60 


ABSORBABILITY OF X-RA DIATION [3 IN AL | 


The results may be stated thus :— 

Every element has its own characteristic secondary 
X-radiations, which it emits when exposed to X-radiations 
of more penetrating type, in agreement with Stokes’s Law 
of Fluorescence. This characteristic line spectrum in X-rays 
determines the variation in intensity of secondary X-radiation 
from the element, the variation in absorption of X-rays in 
the element, the variation in ionization in the element, the 
variation in intensity of corpuscular radiation from the 
element, as the penetrating power of the X-radiaticn is 
varied. 

Thus if we pass a beam of X-rays of gradually increasing 
penetrating power through two substances A and B in the 
gaseous state, then when the X-radiation has not a penetrating 
power close, on the more penetrating side, to that of the 
radiations characteristic of any element in A or B, the 


376 Prof. C. G. Barkla on Typical 


absorption in A is approximately proportional to the absorption 
in B, the ionization in A proporticnal to the ionization in B, 
and the intensity of secondary X-radiation already excited 
in A proportional to the intensity of secondary X-radiation 
already excited in B. But as the penetrating power of the 
primary X-radiation becomes just greater than that of the 
radiation characteristic of an element in A say, this new type 
of X-radiation beyins to be emitted by A, the absorption of 
the primary radiation in A begins to increase, the ionization 
in A begins to increase, the intensity of corpuscular radiation 
from A begins to increase. All these increases occur 
together, and they are, in general, very considerable in 
magnitude. There is every indimaeaa of all these quan- 
tities ultimately settling down to proportionality again with 
the corresponding quantities in B, though in this higher 
ratio. 

The question naturally arises as to the possibility of the 
great increase in ionization being produced not by the direct 
action of the primary rays, but of the secondary ray s—(X or 
corpuscular )—which are ‘connected with the increase in 
ionization. It may easily be shown that the secondary 
X-radiation did not produce more than a very slight increase. 
The effect of the corpuscular radiation will, however, be 
considered as it leads to an interesting result. 

During the transmission of X-rays through a gas, each 
thin layer of gas, unless within about 1 millimetre of the 
boundary in these experiments, is subject to the corpuscular 
radiation from two thick plates of its own substance—one on 
each side. 

Let Xand 2’ be the coefficients of absorption of the primary 
X-radiation and of the secondary corpuscular radiation 
respectively in the gas or vapour, as defined by the equations 


L=lye 7 and), WT, ote 


Let k’ be the coefficient of transformation of X-radiation 
into corpuscular radiation, as defined by the equation 


dK! =k' Ide. 


dk’ being the total energy of the primary radiation of unit 
cross-sectional area transformed into corpuscular radiation 
per second in a layer of depth dw. 

If f, is the fraction of this directed towards, the a eae 
incidence of the primary beam, the total intensity of this 
corpuscular radiation emerging from a thick layer through 


ae " 


| 


—~ 
~] 
~!| 


Cases of Ionization by X-Rays. 
its face of incidence 


-| fade *? ; 


=| fik'le "de ; 
0 
= fikTo) e OFX a 


a0 

if 
le ft ee 
Tk pe ees 


Similarly the intensity of corpuscular radiation from a 
thick layer of gas preceeding from the face of emergence of 
the primary beam is 

i 
! 0 
Fok De an, ’ 
where f) is the fraction directed towards the emergent face, 
and I, is here the intensity of primary radiation emerging 
from the thick layer. 

Hach thin layer of gas is thus exposed to corpuscular 

radiation of total intensity 


BIg fa ee 
d 0 ERE Es. , 
In the cases we are. considering A is negligible in com- 
parison with 2’. 
Therefore each layer of gas is exposed to corpuscular 
radiation of intensity 


0 Ys Si . 
.. Intensity of corpyscular radiation producing ions _ Kk 
Intensity of primary X-radiation producing ions A’" 


If the coefficients of ionization of the two types of radiation 
(primary X and secondary corpuscular) in the gas jtself are 
7, and 2,”, as defined by the equations 

dn=ildx and dn'=1,'\'de, 
where dn and dn’ are the numbers of ions produced in a layer 


of gas of thickness dx by primary and secondary corpuscular 
radiations of intensities I and I’, 


then din!» Kae! 
dn” Wi,’ 

__ lonization by secondary corpuscular radiation 
a lonization by primary X-radiation = 


Phil. Mag. 8. 6. Vol. 20. No. 116. Aug. 1910. 20 


which therefore 


378 Prof, C. G. Barkla on Typical 


Now Mr. Sadler has determined the total ionization pro- 
duced in air by the corpuscular radiation from the incident 
face of plates of various substances compared with the 
ionization produced by the exciting X-radiation ina layer 
of air 1 centimetre thick. Call this Ry. 

The total ionization of corpuscular radiation 


a 12) 
Be rs : 
ae | OM TLDs 
0 
[z,' is the ionization coefficient of corpuscular radiation in air’ | 


fe. 2) 
I —Xq'z 


a mt ; 
= ta ie dx, 


a 
Litt eee ° 
R= = 7 = las 
Na w 
Qe 
pi dipaialg 1 
Sf ikea eS a a pe 
PT Bd ie 
/ Tee dae’ ’ 
n 1G Fa 
— therefore equals 4% Aq ¢ Ry. 
n Pi Lig. 


2 e e e J es s es . 
Now ~ in the case considered is the ratio of ionization in 
1 


r 
air to ionization in air and ethyl bromide, when the corpuscular 
radiation is not excited. 
: J! 
This = — 
4! cS 
+, is the ratio of ionization in air and ethyl bromide to 
1 


approximately. 


that in air, produced by the corpuscular radiation alone. 
This is very approximately equal to 
density of air and C,H,;Br 


density of air ae 


Xz is the absorption coefficient in air of the corpuscular 
radiation from Br (or anything else) excited by the particular 
primary radiation. 

Take the case of ionization by the radiation from Ag. 

Aa =8'8 approximately. 
(See Sadler’s paper on ‘‘ Homogeneous Corpuscular 
Radiation,” Phil. Mag. March 1910.) 

R, is the quantity defined; and by interpolation can be 
shown to be about °8 for Br. For the mixture of air and 
ethyl bromide used in these experiments R, would be less 


than this, as x is less, the active substance Br being diluted. 


Cases of Ionization by X-Rays. 379 


Assuming the absorption of the corpuscular radiation to be 
dependent simply on the quantity of matter passed through, 
R, becomes *3 approximately. 

For /; we will take the value found by Mr. Sadler, 4, as 
approximately correct. 

— therefore =3 x 16x = x 8°§ x ‘3=1°7 approximately. 

We should thus conclude that the ratio of ionization due to 
ethyl bromide to ionization in air at the same pressure would, 
owing to the emission of corpuscular radiation, increase from 30 
to about 80 when using the homogeneous radiation from silver. 

From column 4 of Table I. it will be seen that the increase 
observed in the ionization produced in ethyl bromide when 
_ the ionizing X-radiation was made more penetrating than the 
secondary X-radiation characteristic of bromine was froin 
about 30 to 153 for the radiation from silver. Thus quite a 
considerable portion of the observed increase might be attri- 
buted to the corpuscular radiation. The exact proportion 
cannot be given with certainty. 

On the other hand, Mr. Crowther *, from a study of the 
ionization in ethyl bromide at low pressures, concludes that 
no appreciable portion of the ionization is due to very soft 
secondary radiations, such as these corpuscular radiations. 

It should be observed that we have assumed that the emission 
of corpuscular radiation is not affected by the state of the 
bromine, that is, that the corpuscular radiation is an atomic 
phenomenon simply. Further experiments are being made 
by the writer to test these conclusions. 

The results of experiments may be briefly stated thus :— 

From the results of observations on loniza‘ion in many 
gases and. vapours complete regularity in behaviour has been 
found. Itis necessary and sufficient to know the penetrating 
power of.the characteristic secondary X-radiations emitted 
by the constituent elements to determine the way in which 
the ionization in a gas or vapour varies with the penetrating 
power of the ionizing radiation. (See fig. 2.) In other 
words, it is necessary and sufficient to know the X-ray line 
spectra for the constituent elements. Noanomalous ionization 
by X-rays has been observed. 

A much fuller account of the experiments will be published 
later. 

My thanks are due to Mr. G. H. Martyn, B.Sc., for his 
most valuable assistance throughout the experiments. 

Wheatstone Laboratory, 

King’s College. 


* Roy. Soc. Proc. A. lxxxii. pp. 103-127, 


e804 4 


Cr 


XXXVI. On the Motion of a Particie about a Doublet. 


To the Editors of the Philosophical Magazine. 
GENTLEMEN,— 


N a recent paper* Sir J: J. Thomson investigates the 
motion of an electron about a doublet. I have recently 
been examining this same motion, and, as J have arrived at 
very different conclusions from those of Sir J. J. Thomson, 
a brief statement of my results may not be out of place. 

The equations of motion of this particular problem admit 
of complete integration, providing, I think, the only instance 
in particle dynamics of a soluble problem in which the orbit 
is not confined to one plane. 

As stated by Sit J. J. Thomson, the equations of motion 


are 
d2r és 2Me cos @ 
e. = Ir sin? 6 = i ha 6? = —— “sane Wee 3 ° ° (1) 
_ Me sin 6 
acute oes se ae ae 
a ae 20) 7” sin @ cos Ag? = al Wea (2) 


d : . 
ai? sim o$¢)=0, Sees 


and these have the obvious first integrals of momentum and 


energy, 1 
sin? Od=n, 4, 2) 2 es 
1/324 m2 O21 p2ain® A 42 7 es 
3(7? + 7? 62 + 7? sin? 6 h?) = E+ 2 ee (5) 
Equations (2) and (3) give 


(2 
ae a Me. n2 cos 0 


(1B) Fy (0? 8) = 7? (02 6) = — —* sin 0+ 


leading to the integral 
Me cos 6 n? CG 
m7 Ze 8in? a po) oe (6) 


ROS (op eee aa 
m 


in which € is a constant of integration. From this and (5), 


3 LpPehe ay iao.) 1 oe 
of which the integral is 


C 
ee 


* “On the Theory of Radiation,” Phil. Mag. xx. p. 244. 


~ 


On the Motion of a Particle about a Doublet. 381 


If E is negative the particle falls into the doublet after a 
finite time. The case in which E=0 is exceptional, and 
will be considered later. In all other orbits 7 passes through 
a single minimum value 7)(=,/(C/E)), after which the 
particle passes to infinity, arriving with velocity v(=,/(2E)). 
The changes in 7 are precisely those which occur when a 
particle describes a straight line distant 7» from the origin, 
with a uniform velocity vp. 

The remaining integrals are best obtained in terms of a 
subsidiary variable y given by 


I a aia chink ye) Gay 
t= “tan y, at Rae ieee dy oh G14)) 
Ug 
these equations being mutually consistent with (8). The 


value of dy is rpvodt/r?, and on changing the variable from ¢ 
to y, equation (8) yields the integral 


dé 


x=| Me ne? iJ 
(1 =s Ch cong — 7 Cosee” @) 


which gives a relation of the type 


Me \ 
cos 6=/ {J (aca)xtef. 


and on evaluating @ from (4) the integration is complete. 

The special case of E=U gives motion in a sphere r=7,. 
All the equations of motion are satisfied if r=7) together 
with 


. 2Me n? 
oo Tae é 
is miro’ cos 0 ry sin? 6 
: 4AMe : 
Provided n2< ae there will be two real angles @;, 0, 


for which 6, as given by this equation, vanishes; when n? 
has the critical value, the angles 0; 6, coincide in the angle 
@=tan-! 4/2. There are therefore an infinite number of 
possible spherical orbits for each value of 7,, each orbit being 
confined to a belt of the sphere lying between the cones 4,, 0.. 
Any small departure from spherical motion will be repre- 
sented by giving to E a value slightly different from zero. 
For such an orbit it is clear from equation (8) that r ulti- 
mately becomes infinite or zero, showing that the original 
orbit was unstable. 


O82 Notices respecting New Books. 


Thus it appears that in addition to the circular orbits in 
the cone 6=tan-! 4/2, there are an infinite number of other 
periodic orbits, and also that all these orbits are unstable, 
results which appear to be subversive of the physical theory 
suggested by Sir J. J. Thomson. 

Yours faithfully, 

Cambridge, July 4, 1910. J. H. JEANS. 


——_—_— =. = : 


XXXVI. Notices respecting New Books. 


Bulletin of the Bureaw of Standards. Vol. 6. No. 3. February 
1910, Washington: Government Printing Office. 


Puts part contains three important papers on Radiation : Selec- 

tive Radiation from various solids, by W. W. Coblentz; Lumi- 
nous efficiency of the Firefly, by H. E. lves and W. W. Coblentz ; 
Luminosity and Temperature, by P. G. Nutting. ‘The first is a 
continuation of a detailed examination of the emission curves of 
very various bodies and contains the curves. The second paper 
gives the luminous efficiency of a glow-lamp as ‘43 per cent., while 
that of a firefly is 96°5 per cent. (allowance being made for the 
variation of visual sensibility with the wave-length). The third 
paper discusses the connexion between luminosity and tempe- — 
rature, making use of the author’s visibility function, 


V=V, exp.(—K (A—Aq)’). 


A fourth article is a theoretical and experimental study of the 
Vibration Galvanometer by Frank Wenner; another deals with 
an experimental study of the specific heat of some calcium chloride 
solutions between —35° C. and +20° C.; measured in part by a 
continuous flow method and partly with the use of a Dewar vessel 
containing the brine into which heat is admitted electrically. The 
investigation was undertaken owing tothe extensive use of calcium 
chloride brine as a circulating medium in refrigerating plants. 

The last article is by E. Buckingham on the definition of an 
ideal gas. 'I'o the present reviewer ‘there is something nugatory 
about all such discussions, it being somewhat arbitrary as to what 
the criteria of ideality should be, as soon as we attempt anything 
more scientific than the application of Boyle’s and Charles’ laws. 
But Mr. Buckingham’s article is well worth a detailed consideration 
as it makes clear many things which are often treated very loosely 
in discussions on this subject; and though some of these matters 
have been explained before, it does not appeny to be unnecessary 
to repeat them. 


Be dde -\.] 
XXXVIIL. Intelligence and Miscellaneous Articles. 


To the Editors of the Philosophical Magazine. 


GuNTLEMEN,— 


AGREE with Dr. Kleeman (Phil. Mag. 1910, xx. p. 248) 

- that evidence can be cited from his own work and that of 
others against the view that the y-rays are homogeneous. The 
question, as he points out, is an intricate one, and there exists a 
real inconsistency in the experimental results, different lines of 
work leading to opposite conclusions. The alterations of the pro- 
perties of the y-rays, and the diminution of their absorption- 
coefficients (‘‘ hardening”) by previous screening of the rays, are 
certainly opposed to the view that the rays are homogeneous. 
Dr. Kleeman, however, scarcely does justice to the evidence we 
advanced in Part II. of our paper (Phil. Mag. 1910, xix. p. 725), 
for the belief that the y-rays are homogeneous. According to 
him this evidence depends on the use of a formula ‘‘ which like 
other absorption formule can only approximately represent the 
facts,” and which does not take any account of the production ot 
secondary radiation or of scattering of the primary without change 
of nature. This criticism is singularly at fault. The formula we 
used for the absorption of the y-rays of radium over a semicircular 
are from a point source placed at the centre of a truncated hemi- 
sphere of absorbing metal, as we employed it in our experiments, 
is, unlike other absorption formule, mathematically exact and is 
deduced from the three definite postulates, (1) that the y-rays are 
homogeneous and exponentially absorbed, (2) that scattering of the 
primary radiation does not take place, (3) that no (penetrating) 
secondary radiation is produced in the metal. ‘The formula could 
not hold true if either of the three postulates were false. 
Nevertheless the theoretical formula agreed nearly perfectly with 
the experimental results for the case of lead, with the same value 
for the absorption coefficient as had been found in numerous other 
experiments. For zinc the formula also held for thicknesses 
greater than 2 cm., the results clearly indicating that here a 
penetrating secondary radiation was produced. When the com- 
_ plicated character of the theoretical expressions, involving as 
they do two exponential integral terms, is borne in mind, the 
almost perfect agreement between the theoretical and experimental 
results for lead cannot be dismissed as lightly as Dr. Kleeman 
indicates. 

I take this opportunity of putting right an error arising out of 
a reference we made (p. 730) to some work of Bragg and Madsen. 
We deduced theoretically that the transformation of a fraction, 
varying from zero to unity, of the absorbed primary into a pene- 
trating secondary radiation by different metals would result in 
variations in the ionizations observed with great equivalent thick- 
nesses of different metals in the ratio of rather more than 2:1, 


381 Intelligence and Miscellaneous Articles. 


which agreed with what we had found. We alluded to the fact 
that Bragg and Madsen had obtained a similar 2:1 ratio; but, as 
Professor Bragg has pointed out to me, secondary penetrating 
rays cannot have been the cause of their results, which they ascribe 
to variation in the absorbabilities of (secondary) 6-rays in different 
materials. A part of our variation must be similarly accounted 
for, so that whether secondary penetrating rays had any influence 
remains to be proved. 


Glasgow, July 5th, 1910. FREDERICK Soppy. 


To the Kilitors of the Philosophical Magazine. 


GENTLEMEN,— June 21, 1910. 


In a paper entitled “On the Electrostatic Effect of a Changing 
Magnetic Field” by J. M. Kuehne appearing in the April 1910 
number, the author attributes to me an effort to observe this 
effect described in my paper published in the Physikalische Zeit- 
schrift (vi. p. 474, 1905). In my paper I distinctly disclaimed any 
effort to observe this effect, and pointed out that the experiment 
aimed to show that a plan suggested by Kolacek could not be 
expected to yield any positive results. 

Very truly yours, 
Johns Hopkins University, Joun B. WHITEHEAD. : 
Baltimore, Md. Professor of Applied Electricity. 


To the Editors of the Philosophical Magazine. 
GENTLEMEN,— May 25, 1910. 


In connexion with my note ‘On the Laws regarding the 
- Direction of Thermo-electric Currents enunciated by M. 'Chomas ” 
(Bulletin de la Classe des Sciences of the Académie Royale de Belgique, 
No. 8, p. 903), which appeared in the April number (Phil. Mac. 
xix. p. 508), Professor E. van Aubel, of the University of Ghent, 
has called my attention to a note of his in the “ Chronique et 
Correspondance ” columns of the Paris Revue générale des Sciences 
for December 30th, in which the observations of Jager and 
Diesselhorst are used to disprove the laws promulgated by 
M. Thomas in almost the same way as I use them in my note. 
I regret that I was not acquainted with Professor van Aubel’s 
note when I sent mine to the Physical Society of London on 
January 8th, as I should have been glad to know that the dis- 
agreeable duty of criticising M. Thomas’s theory had already been 
discharged. 
Tam, Yours truly, 
Cares H. Lens. 


SE 


ee ee — 


Curves IV. 


1@%) 
As 
n 


Curves iv. 


' Bos.) os 0-6 


Curves VI. 


Curves VII. 


Phil. Mag. Ser. 6, Vol. 20, Pl. IV. 


oO 
CS EO als | 
[e) ° : } 

A 
ie) Microamberes 
28 21 

24 | 
| 
{ 
\'0 Microamperes 


oHarrock & TYNDALL. 


Curves I. 
POSITIVE DISCHARGE 
Sin 
E.S) Units 
400 = 00 
= X=15 Y=20 
=5:0 
=30 
= 20 
21-5 
=\0 


220 


w 
Is 
Esc oc 


() 
& a : aie a Microamperes 
Curves II. 
iene NEGATIVE DISCHARGE 


Sr Te Ue | (ee 
e aan ual 


Micro amperes 


Curves III. 
ai tn 
E.S. Units. 


Microampetes 
w 


5h, (E.6. Units) 


Curves IV. 


Curves V. 


> Microampberes 


Phil. Mag. Ser. 6, Vol. 20, Pl. IV. 


Curves VI. 


L 
400 


300) 
———t T fa ho! lOnahteho ons 
Curves VII. 
Sain 
E.5) Units 
aaa 
28 21 
a 24 
cl 
SS ae re 


Phil. Mag. Ser. 6,Vol. 20, Pl. V. 


Fie. 7. 


| 
PADIATOR. 


Sw Fay, ange 
eG LE cy SR ee 


Phil. Mag. Ser. 6,Vol. 20, Pl. V. 


BEATTY. 
Fie. 4. Fic. 7. 
i[ | 
Fe RADIATOR. | SW RADIATOR. 
aul | Simm LRAEE detweew plates. a locm, distance between plates. 
] 


Gorvé L Tesla” NAIR 
uw | IL carHope lonisarion iy air | 


70} EH 


4 


1 - 
(ololo} ity iy ou Nace, 
| AIR (ABSC/SSAE x479) 
| 


JonssaTion iN 


60 = 


60 |_ae 


+ 


Fie. 6. 
Ie IL I 


loc curve) 


ABSCISSAE KIO. 


i 
| 


= 
S | 
g S 1 
| 40) 90) y Al 340 + | dL 
z P Sn_RADIBTOR, 2 S 
iS mm. betwean plates. | | 
330 80 a | ie al 
< a 
| L0G. CURVE. 4 
A lonisation \in AIR. 
ee a 70 =H ao i 
2? kc. i—x~—j9 x X 
“1 
10 60 - 10 i | 
g 
[ieee Ss e @ 2 ele 
(00 200 300 400 500 600 700 50 AN = aes LONISATION i Aunt 
i 4 
== — Too 200 300 400 500 600 700 
PRESSURE 114 HAS \b Ss i CS —— 
2 
$ | | 
P40 = me g 
Fra. 5 3 
IG. oO. B 
5 ‘ Frc. 8. 
7 30 ; = x y 
g b 
80 80} 
aa io) 3 JONISATION DUE {2 
d HOMOGENEOUS RA DIAK/ION ALONE. 
70 = 
70 S 
mi Pi Air 
og lr eee < 
p S| 
60 ON/SATION) g 
IN & 
100 200 300 400 500 600 700 8 50) ——_+——;——} 


— PRESSURE IN MMS.— 


CATHODE 
& 


att oF 
ie 


40 


le ae sit 


'YOROGEN| 


in AIR. 


i) 10 20 0 40 50 60 70 80 90 


+ OF ExciTiNG RADIATIONS BY AL 


yoo 200 300 400 500 600 00 
— Pressure IN MM. S— 


THE 
LONDON, EDINBURGH, ayo DUBLIN 
PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


SEPTEMBER. 1910. 


XXXIX. The Consequences of the Corpuscular Hypothesis 
of the y and X Rays, and the.Range of B Rays. By 
W. H. Brace, M.A., F.R.S., Cavendish Professor of 
Physics in the Unwwersity of Leeds*. 


Introduction. 


ia the following pages I have first restated briefly the case 
for the corpuscular hypothesis of the X and y rays. I 
have then attempted to show the consequences to be 

(1) A simple view of the history of the X or y ray. 

(2) The absence of true secondary radiation. 

(3) A true additive principle in radioactive phenomena. 

(4) The absence of specular reflexion. 

(5) The inability of X and y rays to ionize directly ; the 
effect is indirect, the real agents being the secondary 
cathode and 8 rays. 

(6) The general principle that if one radiant entity («, B, 
y, X, or cathode ray) enters an atom, one and only 
one entity emerges, carrying with it nearly all the 
energy of the entering entity. 

(7) A natural division into three groups of the phenomena 
attending the passage of each radiant entity through 
matter. These groups relate to (a) rectilinear move- 
ments during which energy is spent so long as 
ionization is being produced; (6) special encounters 
with atoms on account of which deflexions or scat- 
terings take place without appreciable loss of energy; 
(c) transformations (¥y into 8, cathode into X, &c.). 

* Communicated by the Author. 
Phil. Mag.S8. 6. Vol. 20. No. 117. Sept..1910, 2D 


386 Prof. W. H. Bragg on the Consequences of 


(8) The simple solution of at least two useful ionization 
problems. ‘he second of these leads to a ready 
determination of the relative average ranges of 8 rays 
in various materials (the range being defined as the 
total length of the track when straightened out). 
These fit in very well with results obtained indirectly 
by H. W. Schmidt, and so furnish a general expla- 
nation of the form of the absorption curves of 


B rays. 


THE idea that X and y radiations are both to be regarded 
as consisting of streams of discrete entities has gained 
ground steadily in the last year or two. Sir J. J. Thomson 
looks upon the X ray as a kink in the one tube of force by 
which he represents all the properties of the electron. Ac- 
cording to present knowledge the y ray is of the same nature 
as the X ray, so that an hypothesis regarding the nature of 
the one must be taken to apply to the other also. J. Stark 
has recently developed * the theory, based on the work of 
Planck, that an X ray is a bundle of energy travelling 
without alteration of form. This differs from Thomson’s 
theory in at least one important particular because the latter 
involves a change of form+. J have myself found it con- 
venient to regard the X ray as a negative electron to which 
has been added a quantity of positive electricity which 
neutralizes its charge, but adds little to its mass. 

Whatever view may be taken of the nature of the entity, 
the acceptance of the corpuscle idea modifies our views of 
the phenomena attending the passage of rays through matter, 
and alters the language which we use in describing experi- 
mental results. I think that it leads to a marked gain in 
simplicity, and my object in writing this paper is to show, if 
I can, that this is the case. 

It will be convenient to begin with a brief statement of 
the main arguments for the entity hypothesis, though this 
plan involves some little repetition of similar statements pre- 
viously given. or this purpose it will be best to use the 
results of recent investigations, since they are most fitted to 
serve as a foundation for the case, although I would not 
undervalue the older arguments which first suggested the 
discrete form of the X ray. 

When a pencil of y rays is directed normally upon a thin 
plate, for example a plate of aluminium one or two milli- 
metres in thickness, 8 rays spring out from both sides of the 


* Phys. Zeit. x. p. 902 (1909) ; xi. p. 24 and p. 179 (1910), 
f Phil. Mag. Feb. 1910. 


the Corpuscular Hypothesis of the y and X Rays. 387 


plate *, but very many more are found on the side of the 
plate from which the y rays emerge than on the side through 
which they enter. In fact experiment shows that their dis- 
tribution is just such as should be found if, when they are 
first formed, they simply prolong the line of motion of the 
y rays, and if their subsequent movements are due to the usual 
scattering which 8 rays undergo. Cooksey+ has shown 
that the same lack of symmetry is to be found in the cathode 
radiation which is caused by X rays. 

It is also found that the speed of the 6 ray, which is caused 
by a ¥ ray, is independent of the nature of the atom in which 
it originates, but is directly connected with the quality of 
the y ray. Again the parallel effect is to be observed with 
X rays, as is evident from the work of Dorn, Innes, and 
others, who have made it clear that the speed of the cathode 
‘ rays which originate when X rays fall upon atoms depends 
rather on the nature or quality of the X rays than on the 
kind of atom. But the most accurate and complete proofs 
of these principles have been recently given by Beatty { and 
by Sadler §. 

These facts are of fundamental importance when we come 
to discuss the source from which the 6-ray energy is drawn. 
If it comes from the atom, as was first supposed, we have a 
trigger effect: the y ray is to be considered as precipitating 
an explosion ||. But if this were the case we should expect 
(1) that the direction of motion of the shot, viz. the B ray, 
would have no connexion with the direction of motion of the 
y ray which merely pulled the trigger of the gun; (2) that 
the speed of the @ ray would not depend on the quality 
of the ray, but on some property of the atom corresponding 
to the charge in the gun. The actual conditions are exactly 
the reverse. If we examine the alternative hypothesis, viz. 
that the energy of the @ ray isbrought to it by the y ray, and the 
atom is merely the cause of a transference of energy, we find 
a perfectly satisfactory explanation. The momentum of the 
electron is a persistence of the momentum of the y ray, and 
its energy is derived from the ray ; the electron, therefore, 
continues the line of flight of the y ray with a speed which 
has nothing to do with the atom to which the transformation 
is due, and depends entirely on the quality of the y ray. 


* Bragg and Madsen, Trans. Roy. Soc. of South Australia, Jan. and 
May 1908; also Phil. Mag. May and Dec. 1908. 
+ ‘Nature,’ April 2, 1908. 
t Proc. Camb. Phil. Soc. vol. xy. pt. v. p. 416. 
§ Phil. Mag. March 191v, 
|| ‘ Conduction of Electricity through Gases,’ 2nd ed. p. 320, 
212 


388 . Prof. W. H. Bragg on the Consequences of 


We therefore conclude that the energy of the £ ray is 
derived from that of the y ray, and similarly the energy of 
the cathode ray from that of the X ray. We are then ina 
position to take into account another experimental result. 
The velocity of the cathode particle ejected by the X ray is 
found to be the same, or nearly the same, as that of the 
cathode particle in the original X-ray tube. There is no 
doubt as to the approximate truth of this statement, though 
accurate experiment is wanting. Now there can be no question 
of the storage of X-ray energy in an atom until there is 
enough to provide for the ejection of a cathode ray, for then 
the nature of the atom would again be of influence, and we 
should revert to all our previous difficulties. One X ray 
must be enough to provide one cathode ray. Nor does it 
seem possible to suppose that the energy of several cathode 
particles can be stored up in an atom until there is enough 
to produce one X ray; for amongst other considerations there 
would then be no apparent reason why the speed of the cathode 
ray should influenee the quality of the X ray so directly as 
it does. Hence the X ray cannot have more energy than 
was possessed by the cathode particle in the X ray bulb. 
Put the two statements together and we find that one cathode 
ray impinging on an atom may produce one X-ray and no 
more, and in its turn the X ray through impact on an atom 
(not necessarily the first it meets) produces one cathode ray 
and no more, handing on its energy and its direction of 
motion. 

It is this conclusion which seems fatal to the spreading 

pulse theory. The latter taught us that when an electron 
was arrested the energy set free travelled out in all directions 
through space on an ever enlarging surface. We now find 
that we must have the energy of the X ray confined within 
very narrow bounds which are not to widen as the X ray 
travels, so that when at last the transference of energy takes 
place the energy is all in one spot ready for the sudden 
change. The speed of the cathode ray caused by the X ray 
is the same no matter where it comes into being. We cannot 
allow the energy of the X ray to spread even a little. The 
ray is to be considered as a minute entity of some sort, its 
energy as it travels being always bound up in an unaltering 
volume of atomic magnitude at the most. 
’ This isa brief statement of the case for the entity hypo- 
thesis, containing only one main line of argument. Many 
subsidiary considerations are omitted. It is worth observing 
that it turns on questions regarding energy. 

We must of course ask what we lose by the adoption of 


the Corpuscular Hypothesis of the y and X Rays. 389 


the new hypothesis, with the consequent abandonment of the 
spreading pulse theory. Only one thing of value: viz. the 
easy explanation of the partial polarization of a primary beam 
of X rays, and of the more complete polarization of secondary 
beams. Those who would maintain that the entity contains 
a wave-motion within it might argue that there is no loss of 
this kind; but such a position. would seem unsound until 
there is a clear expression of the meaning and properties of 
an entity or bundle of energy with a wave-motion inside it. 
It is to be observed that the polarization of light is a very 
complex phenomenon which is capable of the closest exami- 
nation, and that the undulatory theory of light explains it 
with great exactness. It is possible to overrate the im- 
portance of the ability of the pulse theory to explain the 
polarization of X rays, because it may be imagined that in 
this case also a complex effect is successfully accounted for. 
As a matter of fact the polarization of X rays is quite a 
‘simple effect and bears but a meagre resemblance to the 
polarization of light; there are, for example, none of the 
elaborate and beautiful effects of crystals. The polarization 
of the X ray consists only in the fact that if it is deflected it 
is more liable to move in one particular plane passing through 
its line of flight than in another: a billiard ball with side on 
does as much, or more exactly still, a spinning tennis ball. 


If we accept the entity hypothesis the processes of the 
X-ray tube assume a new aspect. We gain in precision of 
statement and in clearness-of outlook. The stream:of cathode 
rays is directed against the anticathode; we no longer say, 
somewhat vaguely, that part of the energy goes in heat, part 
in secondary cathode radiation, part in X rays. We must 
not imagine a cathode ray to ricochet hither and thither 
among the atoms of the anticathode radiating X-ray energy 
at every turn. No doubt it does so radiate some energy, but 
the amount is trifling, and has nothing to do with X rays. 
We must rather say, that when each cathode particle strikes 
the anticathode it may fritter away its energy into a form 
which finally takes that of heat, or it may be splashed back 
against the glass wall of the tube, and cause phosphorescence 
and other etteets, or, again, it may disappear (not necessarily 
at its first meeting with an atom, nor before it has spent any 
of its energy), and the complete disappearance of the cathode 
ray as such will then be simultaneous with the production of 
the X-ray entity. In the last case the entity starts off on 
its straight line course endowed with a penetration which 
the cathode ray did not possess.. When it meets an atom, 


390 Prof. W. H. Bragg on the Consequences of 


there is an overwhelming probability that it will go through 
without effect; but it may be deflected, and again it may in 
its turn be replaced by a cathode ray like the original one. 
We may think of the whole affair as the history of a small 
quantity of energy carried first in the X-ray bulb by a 
cathede ray, transtormed into the energy of an X ray, with 
perhaps further reconversions ; frittered away while it takes 
the cathode ray form, carried intact while it has the X-ray 
form, until finally it has all been spent. 

It is never reinforced at any stage of its journey, for there 
is no unlocking of the internal stores of atomic energy, ac- 
cording to the most recent experimental evidence. Both 
Bumstead and Angerer, working independently, found there 
was no trace of a difference in the amount of heat generated . 
by a stream of X rays in two different metals, such as would 
be expected if any part of the heat were due to atomic 
energy set free by the X rays. Moreover, no arrangement 
of screens or reflectors about a stream of X or y rays causes 
any increase in the total ionization produced by the stream, 
so far as we have been able to discover. It is only possible 
to increase it in one place at the expense of a decrease in 
another. In this sense at least there is no such thing as 
‘secondary radiation.”’ 

The term “secondary radiation” is largely used, and is 
often quite satisfactory; but it may have many meanings 
not all of which are true to fact. It is convenient for the 
time to continue the argument of this paper in the form of 
a discussion of the circumstances under which the use of the 
term is justified. For it js obvious that as long as we retain 
the idea that secondary radiations may add themselves to 
primaries, tertiaries to secondaries, and so on, we are op- 
pressed with the sense of a complexity which must add 
greatly to our difficulties. If, on the other hand, we can 
permit ourselves to think that there is no indiscriminate 
addition of this kind, but that the appearance of each indi- 
vidual secondary entity is marked by the simultaneous dis- 
appearance of a primary entity; further, that the secondary 
inherits the energy of the primary, and, in some cases, its 
direction of motion; and further still, that the secondary can 
for all practical purposes be looked upon as a continuation 
of the primary, sometimes modified in form, then we obtain 
a simplification worth having. Let us, therefore, consider 
the matter a little more in detail. 

When the electron, as a 8 or a cathode ray, dives into an 
atom and is thereby deflected, as is occasionally the ease, the 
electron moves off in a new direction, but it can hardly be 


the Corpuscular Hypothesis of the yand X Rays. 391 


ealled a new ray. We may call it a secondary ray if we 
please, but we may just as well say that every molecule of a 
gas is a primary molecule before a collision, a secondary 
afterwards, a tertiary after two collisions, and so on ; and it 
would be worse than useless to do so. Again, when Geiger 
shows that an & particle may be deflected or scattered he does 
not speak of a secondary aray. When an X ray entity is 
transformed by an atom’s action into a cathode ray, or a 
y ray intoa B ray, we may speak of the new rays as secondary 
rays, and now the term is really convenient ; but it must not 
be taken to mean too much. There is a change of form of 
the entity, and that is all. When an X ray entity is deflected 
in passing through an atom, or is “scattered” in the usual 
phrase, the term secondary radiation is really inappropriate, 
because it is but the X ray entity swinging off in a new 
direction. Barkla has shown that when primary X radiation 
falls upon any metal (from Cr to Ag at least), so long as the 
penetrating powers of the primary exceed a certain limit 
peculiar to that metal, a homogeneous X radiation is emitted 
which is characteristic of the metal, and is less penetrating 
than the primary. Here the term secondary would seem to 
have a real meaning, for we wish to describe the fact that 
a primary X ray entity possessing energy of any amount 
above a certain minimum is replaced by a secondary X ray 
entily possessing an energy characteristic of the particular 
metal, and always less than that of the primary. The effect 
is simple enough to be described in this way, for energy con- 
siderations show that it can only be a case of one entity 
replacing another, not of two or three replacing one, nor of 
one being added to the original. It is not clear, however, 
that a transformation of this kind actually occurs, a trans- 
formation, that is to say, which makes the primary differ so 
much from the secondary that a real difference is to be re- 
cognized by the use of different terms. I hope to be able to 
show later that there are good grounds for presuming a 
double transformation, the first stage being a conversion of 
the primary X ray-into a cathode ray stage, during which a 
loss of energy occurs, and the second a reconversion into the 
X-ray form. In any case it is enough for the present that 
the secondary must draw its energy from the primary, and 
the appearance of the former implies the disappearance 
of the latter. 

There is another case which must be considered specially. 
McClelland * has explained certain of his experiments on the 


.* Proc. Roy, Soc. Ixxx. p. 601, (1908). 


_———_——————SS eee 
Se a 


Se ee 


so SSS 


ee SSS 


Seg SE 


392 Prof. W. H. Bragg on the Consequences of 


scattering of 6 rays by supposing a real secondary radiation 
to be added to a reflected primary. The experiments are 
simple. When a stream of 8 rays falls at an angle of, say, 
45° upon an aluminium plate, it is found that the @ rays 
which leave the plate on the incidence side are not distri- 
buted symmetrically about a normal to the plate, but show a 
maximum -in a direction which is separated by the normal 
from that of the incident stream. When the plate is of lead 
or any other substance having a high atomic weight, the 
effect is much less marked. In fact it looks as if there quae 
a contused specular reflexion at the surface of the plate 
coupled with a radiation scattered in all directions. McClel- 
land therefore divides the scattered rays into two groups, the 
first of which consists of 8 rays from the primary stream 
reflected by the surface of the plate like light by a mirror, 
the second of a set of true secondary rays. 

Let us first consider the question of specular reflexion. 
All the evidence we have regarding the actions and reactions 
between atoms and radiant entities shows that each atom 
when in collision with an entity has to bear the shock alone: 
it receives no support from its neighbours, even when they 
form parts of the same molecule, a fortiora when they are 
only neighbouring atoms in the surface of a plate such as 
McClelland used. It is this which makes radioactive 
measurements independent of physical and chemical con- 
ditions. The point seems to be firmly established now, for 
though at times evidence has been brought forward which 
has at first appeared to contradict the principle, more careful 
examination has always shown the evidence to have been 
mistaken. The principle may be expressed by the statement 
that the action of a molecule on one of the radiant entities is 
the sum of the effects of the actions of its component atoms, 
no allowance for constitutive influences being necessary. 
One or two examples will] be sufficient. 

The stopping power of a molecule for « rays is the sum of 
the stopping powers of the individual atoms of the molecule. 
During 1908 I measured as carefully as possible the stopping 
powers of a number of gases which were prepared in a 
very pure state by Dr. Rennie and Dr. Cooke of the Adelaide 
University. The range of the « particle can be measured to 
much less than one per cent. The additive principle was 
found to be true within the errors of experiment; both for 
stopping powers measured with respect to RaC, and those 
measured with respect to the « particles of Ra A. The two 
sets are not quite the same”. 


_* Bragg, Phil. Mag. Apvil and Sept. 1907. 


the Corpuscular Hypothesis of the y and X Rays. 393 


Again, the absorption and scattering coefticients of liquids 
and compounds for 8 rays have lately been the subject of 
careful measurement by Schmidt* in Giessen, and by 
Borodowsky ¢ in Manchester, and the additive principle was 
fully confirmed in this case also. 

A radiant entity, therefore, acts on one atom at a time; 
and if its direction of motion is altered by a collision, the 
alteration is determined by the mutual relations of the entity 
and the atom alone. Neighbouring atoms have nothing to 
do with it, and it is quite immaterial whether or no there is 
a surface close by which separates one lot of atoms from 
another. On the other hand, specular reflexion, such as the 
reflexion of light in a mirror, depends on the conjoint action 
of the atoms of the reflecting surface. It cannot be supposed, 
therefore, that one part of the scattered @ radiation examined 
by McClelland consists of rays reflected like light: and this 
being so, it is probable that the description of the remainder 
as a true secondary is wrong also. In fact there is a much 
more direct explanation of the whole effect. 

When an entity passes into an atom there is a chance of 
deflexion through any given angle. Radii may be drawn 
from the atom, each representing by its length the chance of 
deflexion into the direction in which it is drawn. The 
extremities of these radii will lie on a surface the form of 
which will represent graphically the probable results of the 
encounter ; and its form will vary with the atom, with the 
nature of the entity, its speed, and so forth. Asa rule the 
lighter the atom the more eccentric is the oval surface. The 
surface is one of revolution, the axis being the original line 
of motion of the entity. A section through the axis will 
therefore express all there is to express ; and such a section 
may be called a ‘‘deflexion oval.” It must be one of the 
objects of experiment to determine the forms of the deflexion 
ovals in all possible cases, for clearly, until we know the 
probable results of an encounter between a given entity and 
a given atom, we cannot calculate the result of the attempt 
of an entity to pass-through a plate which is an aggregate 
of many atoms; in other words, we are not in a position to 
calculate with safety the absorption coefficients or reflexion 
coefficients of 8 rays. Although we do not yet know the 
exact form of the oval when a @ ray impinges on an atom, 
we do know that it is far more eccentric for an aluininium 
atom than for a lead one. The heavy atom is much more 
likely to swing round the electron than the light one; when 
. * Phys. Zeit, xi, p. 262 (1910). 

T Phil, May. April 1910. 


394 Prof. W. H. Bragg on the Consequences of 


a stream of @ rays falls upon a lead plate far more are turned 
back than when the plate is made of aluminium. 

Suppose that @ rays fall on an aluminium plate, as in the 
figure. Suppose one of the rays to be scattered by some 
atom in the plate at P. The chances of deflexion in various 


Ries i 


directions are represented by the radii of the deflexion oval, 
which is roughly drawn as a dotted line. The chances of 
emergence have now to be taken into account ; the deflected 
ray has less chance of getting out the more parallel is its line 
of movement to the surface. Hach radius of the oval must 
be multiplied by a factor approaching the form e—4sec9 where 
dis the depth of the atom below the surface and @ is the 
inclination of the radius to the surface normal. The ends of 
the radii thus obtained lie on a new surface which is similar 
to McClelland’s ; its section is indicated by the firm curved 
line in the figure. It isin the right sense asymmetrical with 
respect to the normal; and the asymmetry is greater for 
light atoms than for heavy, because the lighter the atom the 
more eccentric is the oval. Thus McClelland’s results are 
explained without the necessity of introducing the two hypo- 
theses of specular reflexion and true secondary radiation with 
all the complexities they bring in their train. 

Yet there is one way in which a sort of secondary 8 radiation 
might occur. Canan electron in flight so collide with another 
as to give it a large share of its energy, so that one # ray is 
replaced by two of much less penetrating power? There 
is no obligation to think so at present ; but the case is worth 
considering, for it simplifies matters very much if we can 
conclude that no such obligation is likely to arise. There 1s 
however, not much to guide us. We may to some extent 
argue from the behaviour of other entities. An & particle 


the Corpuscular Hypothesis of the y and X Rays. 395 


has a considerable speed, say 2 x 10° cm./sec., and as an atom 
of helium it must contain several electrons, yet we never find 
in the gas traversed any electrons moving with a speed of 
more than a few volts: that is to say, we find only 6 rays. 
Again, one aray never gives rise to two «rays: nor one 
X ray to two X rays, so far as we can see. ‘The enquiry 
really resolves itself into the difficult question of the way 
in which ionization comes about. There are indications 
that it is not a straightforward process in which the moving 
entity drives out the electron from the atom by direct collision, 
because, in the case of the @ particle at least, the energy 
spent is not always proportional to the ionization produceed— 
there must be an intervening link; and because, as already 
said, the ejected electrons all seem to have speeds of the same 
low order. ‘There is indeed little certain information on these 
points, and it can only be said that to all appearances ionization 
is the result of the passage of entities through molecules, and 
that the observed facts can be expressed on the simple hypo- 
thesis that there is a gradual drain on the energy of the 
entity but no large change at any one encounter with an 
atom. Of course it may well be asked, what then does 
happen when one electron moves so directly upon another 
that we may expect a collision such as occurs when one 
billiard ball strikes another? But then we have here pre- 
conceived ideas of volumes, surface contacts, and elasticities, 
which we must not carry over to the case of electrons 
encountering each other. There is really nothing to compel 
us to handle such electrons as anything more than mere 
centres of force: if we give them dimensions, it is only to 
make them have the right amount of electromagnetic mass. 
Even when we take this view we have no sure ground on 
which to base a calculation as to the probable result of an 
encounter, because the electrons in the atom cannot be con- 
sidered separately ; each one is backed up by unknown 
linkages with positive electricity and with the general frame- 
work of the atom, as we know from the fact that the scattering 
of 8 rays depends verv greatly on the atomic weight of the 
scattering material. To sum up, there is nothing to be said 
against, and something to be said in favour of, the simple 
hypothesis that the 8 particle gradually spends energy along 
its track, but does not lose any material portion of its energy 
on account of the violent deflexions to which it is frequently 
subjected. Its career is like that of an @ particle with many 
more deflexions in it, though there is nothing at present to 
prove that if the track were straightened out the length of it 
would be constant, as in the case of the larger entity, the 


596 Prof. W. H. Bragg on the Consequences of 


range of which can be found with precision. We may think 
of the 8 particle as possessing an average range in a given 
material, best expressed perhaps as a weight of material 
crossed. For purposes of definition we may suppose the 
track to be the axis of a cylinder of a small cross-section s ; 
then if ds is the weight of the cylinder, d is the range. 
I hope to be able to show presently that it is possible to find 
the relative values of d for given @ rays in various substances. 

We have already sutticient information to give us some 
idea of the lengths of the short portions which make up the 
total range. ‘The work of Madsen* shows that such @ par- 
ticles as have been turned aside from a main stream passing 
normally through an aluminium sheet ‘004 em. thick are not 
likely to experience a second deflexion in the same plate. 
Thus 8 particles of a speed approximating to that of light 
must often go through a tenth of a millimetre of aluminium 
without deflexion, or through the equivalent 20 em. of air. 
Similar conclusions may be drawn from an earlier paper by 
Crowther tf. Crowther does indeed state that the scattering 
of a pencil of 8 rays is complete when it has passed through 
"015 cm. of aluminium ; but he uses- the term in a special 
sense relating to the details of his experiment. It does not 
mean that after going through such a plate the stream of 
8 rays has lost all sense of direction, and the various rays 
are heading every way; for his figures show that 30 per cent. 
of the rays which emerge from the plate and were originally 
directed normally upon it retain so much of their original 
direction as to be grouped about the emergent normal in a 
cone of a semi-vertical angle between 4° and 5°. The solid 
angle of such a cone is about 54, of that of a hemisphere. 

I have now considered one by one several possible causes 
of complexity; and I would conclude that on the whole they 
can be put aside as having at present no obvious existence. 
In this way we arrive at a comparatively simple idea of the 
history of the radiant entity whatever its kind, a, 8, y, X, or 
cathode ray. In each case there is an initial store of energy 
communicated to the entity : the subsequent motion is recti- 
linear, varied by encounters which change the direction of 
the motion but not its energy: ionization, if it takes place 
at all, takes place along the track ; and it is in this way that 
the energy is drawn upon. The form of the entity may 
change, y into 8, X into cathode ray, and so on; but there 
is so little change in anything but form that practically we 
may assume a continuity of existence. 


* Phil. Mag. Dec. 1909. 
+ Proc. Roy. Soc. March 1908, Ixxx. p. 186. 


the Corpuscular [Hypothesis of the y and X Rays. 397 


There are therefore three main subjects of measurement in 
respect to each entity: (a) the expenditure of energy along 
the path, (b) the form of the deflexion oval, (c) the chance 
of conversion of form. Let us consider to what extent these 
measurements have ,been made, and also some methods of 
making them. 

Let us take the « particle first. The case is an especially 
simple one because there is no conversion of form, and very 
little chance of deflexion until the speed has greatly diminished 
and the range is nearly completed. Hence the particle’s 
properties are almost entirely expressed when its range is 
determined; and this has been done with some thoroughness. 
The feeble but very interesting deflexions which do take place 
have been measured by Geiger. Our knowledge of the 
a particle is fairly complete in the sense that we know what 
to expect when any given screen is placed in the path of any 
given stream of radiation. We may go on to consider some 
of the other radiations of which we know less. 

The X and y rays have also their special points of simplicity, 
but they form an almost exact antithesis to the « rays. Here 
it appears that the expenditure of energy along the track is 
either negligible or non-existent. The rays do not ionize 
directly. Nor is the deflexion oval a very important thing. 
The most important feature is the chance of transmutation 
of form, the X ray being sometimes replaced by a cathode 
ray, the y by a B ray. 

‘he argument that the X or y entity spends no energy 
along its track arises simply from the fact that it produces a 
cathode or a 8 ray of the same speed, no matter how much 
material it has already traversed. It cannot keep its energy 
intact while traversing matter and at the same time cause 
ionization which involves the expenditure of energy. (Gases 
which are crossed by X and y rays are ionized, but that is 
because they produce cathode and 8 rays respectively: and 
these latter do the work. Of course it may be said that the 
conversion of one X ray into one cathode ray is ionization: 
and so it is ; butit is natural to keep this solitary and peculiar 
event distinct from the general ionization of the gas along 
the track of an entity. 

This deduction seems to afford an opportunity for putting 
our hypothesis to the proof. What experiments have been 
made from which we may determine whether or no X and 
y rays lonize gases directly ? 

McLennan describes an experiment (Phil. Mag. Dec. 1907) 
in which he shot y rays through two ionization-chambers, 
one made of lead, the other of lead lined with aluminium, 


398 Prof. W. H. Bragg on the Consequences of 


and compared the ionization current in the two cases. He 
supposed that the ionization could be assigned to two sources, 
one the direct action of the y rays on the gas, the other the 
secondary rays caused by the y rays to issue from the metal 
sides of the chamber. ‘The former would be the same for 
the two chambers, let it be called Ip: the second would not, 
let it be Igy, for the chamber which is all lead, and Ig, for the 
one which is lined with aluminium. He then assumed that 
Ig, ='286 x Isr, since Eve had shown that when y rays fell 
on lead and aluminium plates the returned 8 rays were in the 
proportion of 100 to 28°6. 
Thus : 


Ip +Isr= 90°05 (total ionization in the lead chamber). 
Ip +Ig4=49°5 (the aluminium lining having been inserted). 
Tey ="286 x Igy 


Hence he found that Ip=33'05; Is, =57 00, In, =16:35 
and concluded that Ip, that is to say the result of the direct 
action of the y rays upon the gas, was very considerable. 

The source of error in this calculation is the assumption 
that Is,=°286xXIcgz. It was not known at that time that 
this relation only holds in respect to the @ radiations from 
the front face on which y rays fall: the 8 radiations which 
issue from the face of a plate from which y rays are emerging 
may even be greater tor aluminium ‘than for lead: and 
McLennan’s results depended on both incidence and emer- 
gence rays. It was not right to use Eve’s figures, which 
referred to a special case of incidence rays; and there is no 
contradiction ot the deduction we have drawn from the entity 
hypothesis, viz. that Ip is zero. 

Again, W. Wilson records * measurements of the ionization 
in an electroscope made partly vf aluminium and partly of 
brass when the pressure of the air was varied from one to 
forty atmospheres: the yrays came from RaC. He supposes 
that “the total ionization due to the secondary 8 rays at 
different pressures will be given by B(1—e~*””) where B is a 
constant, p the pressure and 2 the coefficient of absorption,” 
and further that “the ionization due to the y rays will be 
given by a term of the form Ap, where A is aconstant.” He 
then finds that B must be 6°6 times A, and that the ionization 
due to the secondary rays is therefore several times the 
ionization due to the direct action of the y rays on the gas. 
This is of course nearer than McLennan’s result to what we 
now expect, but it still ascribes some effect to the direct 


* Phil. Mag. Jan. 1909. 


the Corpuscular Hypothesis of the y and X Rays. 399 


action of the y rays. The fact is, however, that the division 
of the ionization into these two terms is not quite right, even 
supposing the ionization due to the y rays to include the 
lonization due to the B rays generated by the y rays in 
the gas. 

Let us consider so far as we can what should be the amount 
of ionization in a gas through which y rays are passing, 
assuming the entity hypothesis and its consequences. There 
are two cases at least in which the solution is fairly easy and 
satisfactory. The easier one is the case of an ionization 
vessel lined completely with any material, provided only that 
it is so thick that 8 rays cannot cross it. The other is the 
case of a large but shallow ionization vessel, the top and 
bottom of which consist of two paralle! plates, one of which 
is made of a substance having about the same atomic weight 
as the air which the vessel contains. Let us take the latter 
case first. 

It simplifies considerations of this kind to remember that 
the spacing of atoms plays a subordinate part in them. 
Suppose, for example, that a stream of 6 rays passes up 
normally to a plate through an opening in it at A, and that 


Fig, 2. 
ee ee 
Cnn earner nn corene i neeeemeneene ne e 
Bonanno nnn eee nen nenscecateneentnentnneln B 


B, C, and D are imaginary surfaces in the air parallel to the 
plate. The 8 rays cause a certain ionization in the air 
between the planes Band (. It would make no difference 
in this amount if the air between C and D were compressed 
into a thin layer lying along C or indeed anywhere above it, 
so long as the air between B and C remained in a uniform 
layer between and parallel to B and C. It would be the 
same even if air were brought down from above C and laid 
in a layer along C in such quantities that no 8 rays could 
get through it; or if a plate composed of atoms of nearly the 


400 Prof. W. H. Bragg on the Consequences of 


same weight as the air atoms were placed along C. If the 
distances of the planes from A were b,c, and d, and if we. 
might assume the 8 rays to be spent exponentially with a 
space coefficient A, the ionization between the planes B and 
C would be I(e~*”—e-) in all the cases just described, 
I being the initial energy of the radiation as it comes through 
the hole at A. There is no need to trouble about secondary 
radiation from a plate at C containing light atoms only, even 
though we know that atoms of carbon and oxygen can return 
some § rays: all such effects are already fully accounted for 
in the formula. 

Consider now a stream of y rays passing normally upwards 
through the lower plate bounding such an ionization-chamber. 
The upper plate can be made of cardboard, or some material 
having approximately the same average atomic weight as air. 

Let & be the absorption coefficient of the material of the 
plate for y rays in the sense that rays of energy I lose an 
amount of energy kIdw in passing normally through a sheet 
weighing dx grams per sq. cm.: # is then the weight absorp- 
tion coetticient. The meaning of this is to be that the energy 
kIda becomes energy of @ rays which at the start continue 
the line of motion of the y rays. 

Let X be the similar coefficient of the plate for 8 rays such 
as these y rays produce. This means that when a layer of 
the same material as the plate, weighing w grams per sq. em. 
is placed normally to a stream of @ rays of energy I, the 
energy which gets through the plate and is spent in ionizing 
the air on the other side is Ie~**. It is worth observing that 
if some other gas, say a heavy one like methyl iodide, were 
substituted for the air, the gas would return more of the 
radiation into the plate, so that more would be spent in the 
plate and less in the gas: it might be said that the absorption 
of a plate depended on the gas or other material above it. 

Let k! and X! be the corresponding coefficients for the y 
and @ rays in air. 

The @ rays originated in a layer of weight dx, which is at 
such a distance down in the plate that a layer of weight « 
lies above it, will have an energy kle“dz, where I is the 
energy which the y rays possess as they enter the ionization- 
chamber. These 8 rays at first move directly upwards 
towards the chamber, and a certain fraction, viz. e~**, of 
their energy is transmitted across the layer « into the 
ionization-chamber. The whole energy emerging is therefore 


t 
kle#-*"dz; and if the plate is thick enough to stop all 
a rays we may put the thickness ¢ equal to infinity. The 


the Corpuscular Hypothesis of the y and X Rays. 401 


emerging energy of 8 rays is therefore kI/(AX—&), which may 
practically be simplified to £I/A, since & is usually so small 
compared with 2. 

The ionization produced in the ionization-chamber may 
be taken as kI(1—e-*?)/X, where D is the depth of the 
chamber multiplied by the density of the air. This is not 
strictly correct, because some of the 8 rays will strike 
against the side walls, which we cannot do without, and will 
not spend se much energy in the layer of air (of weight D) 
as they ought to do according to the definition of X/. If we 
had a material which threw back all 8 rays completely, we 
could avoid this error; but there is nosuch material. It can 
be lessened by having a wide and shallow chamber. ‘There is 
an error of a different nature in that X' was defined with 
reference to rays striking a layer of air normally, whereas 
the 8 rays emerging from the plate will be moving in all 
directions. But it is not worth while to attempt to avoid 
such errors just now: it is probably a still greater error to 
have assumed an exponential law, and our object is to obtain 
a theoretical result accurate enough to tell us what we should 
look for. : 

We have now to take into account the ionization due to the 
8 rays produced by the y rays in the air of the chamber. 
This may be done by direct calculation, or in the following 
way which seems interesting. 

lf the plate which forms the base of the chamber were 
replaced by a plate of nearly the same atomic weight as 
the air in the chamber, the y rays would then pass through 
the same sort of atoms throughout their course. Considering 
a short path of the course in which there is no great absorption 
of the y rays, strata of equal weight convert equal quantities 
of y-ray energy into B-ray energy, and will show equal 
ionization even though the ionization in any stratum is not 
wholly due to the 6 rays made in that stratum. The energv 
spent on ionization in any stratum is practically equal to the 
y-ray energy converted in that stratum: thus the ionization 
in this particular ionization-chamber is measured by k'DI: 
the X' does not come in. If we now replace the bottom plate 
of constants k’ and 2’ by the plate of constants s and A, we 
add a source of ionization amounting to kIl(1—e-*P)/a, but 
we take away a source of ionization amounting, by the same 
rule, to k’I(1—e-*?)/x’. We also provide a plate which 
turns back more effectively some of the 8 rays made in the air 
of the chamber and in the plate at the top, but these are not 
many and we may neglect them. Thus the ionization in the 


Phil. Mag. 8. 6. Vol. 20, No. 117. Sept. 1910. 248 


4()2 Prof. W. H. Bragg on the Consequences of 


chamber is expressed by 


' te ne 
T4 Dal + (5 1) =e yi 


Ii k/A=k'/n' the expression becomes [Dk’ simply: and the 
relation between ionization and pressure, measured by D, 
becomes a linear equation. If k’/d’ is greater than k/A the 
curve is convex to the pressure axis, and if less it is concave. 
So far as I know, no experiments have ever been carried out 
with an ionization-chamber of this form in which y rays have 
been employed to ionize air at different pressures. In the 
experiments of Kaye and Laby * the ionization-chamber was 
wholly made of one metal aluminium : in those of W. Wilsont 
it was partly of brass and partly of aluminium. Ifthe y rays 
have been hardened by a lead screen, k and k’ are nearly 
equal, in fact the absorption coefficients of a number of sub- 
stances are nearly the same. Now the 8 ray absorption 
coefficients are somewhat smaller for light atoms than for 
heavy, so that &/A is less than k’/d’ and the curve, in the 
case I have considered, should be slightly convex to the 
pressure axis. When the top and bottom plates are both of 
aluminium, it should be slightly concave, as will be shown 
presently: Kaye and Laby found this to be the case. 

In the case of y rays, X’D is generally small, unless the 
pressure of the air in the chamber is very great: the 
expression then becomes 


DIN ep pile: cs de Ee 

| 1} r la i 2 i 
There is a term in this expression which is proportional to D 
and therefore to the pressure, but it does not represent 
exactly the aetion of the y rays on the air, as some have 
supposed. Nor does it represent the action of the secondary 
rays from the walls entirely. And again it has sometimes 
been stated that a term proportional to the square of the 
pressure will be required to represent the ionization due to 
the 8 rays made by the y rays in the gas. Clearly this is 
not quite true. 

In the case of X rays k is usually so much greater than k’ 
that the latter may be neglected, and d’J) is so large that e~*? 
is negligible also. The exponential term is only to be retained 
when the pressure of the gas is so low that the cathode rays 
originating in the walls of the chamber can get across it in 


* Phil Mage. Dec. 1908. 
t Phil. Mag. Jan. 1909, 


the Corpuscular Hypothesis of the y and X Rays. 403 


appreciable quantities. At ordinary pressures the formula 
becomes I( Dk’ + k/A). 

In this form it may be tested experimentally. It may be 
well to repeat that this formula is deduced on the suppo- 
sition that X rays do not ionize a gas directly, but indi- 
rectly through the intermediate action of the cathode rays 
produced by the X rays in the metal through which they 
enter and in the gas which they cross. The term Ik/A 
represents the effect due to the cathode rays from the metal ; 
IDk’ represents the effect due to the cathode rays formed in 
the gas. The first of these can be determined by experiment 
in a given case; the second can he calculated from the first 
when measurements have been made of k/k’, A, and D. If 
then the ionization produced by the X rays in the gas 
(directly or indirectly) is also found experimentally, it can 
be seen whether the calculated indirect effect is sufficient to 
account for it all, or whether there is something left over which 
must be ascribed to the direct action of the X rays. 

I have made a number of experiments of this kind and 
have found that the results were always to be explained on 
the supposition that there was no direct action of the X rays. 
An example will show the usual extent of the agreement. 

An ionization-chamber was made of brass, lined with 
aluminium to avoid disturbances due to the secondary X rays 
of brass, and again with paper to cut out the secondary cathode 
rays from the aluminium. The chamber was cylindrical, 
3°6 cm. deep and 10 cm. in diameter. A pencil of primary 
X rays was passed in along the axis through an opening 
1 cm. in diameter. When a card was placed over the 
opening, and nine thicknesses of silver-foil placed on the 
card on the side next the ionization-chamber, the current 
was 150°0 on an arbitrary scale: when the foils were placed 
the other side of the card the current was 70°3. The difference 
79°7 was due to the cathode rays from the silver: 7. e. we 
may take Ik/X to be 79°7. The absorption coefficient k was 
then found by placing various thicknesses of silver under the 
card, and measuring. the current in each case. The curve 
obtained when the results were plotted was not far from 
exponential, and gave k equal to 43:2 for the primary rays 
after passing through 9 foils. The absorption coefficient 
required is that which measures the conversion into cathode 
ray energy, excluding secondary X rays. It is therefore 
better to put the absorbing sheet close to the ionization- 
chamber so that secondary X rays may be taken in, though 
there is still some error due to the difference in quality of the 
primary and secondary rays. The absorption coefficient for 

2H 2 


AOA Prof. W. H. Bragg on the Consequences of 


the 8 rays in silver was found by placing one, two, four, 
eight, and twelve silver foils on the side of the card next the 
chamber and observing the gradual rise in the cathode ray 
effect: this gave X equal to 3550. The quantity * was not 
found directly. The absorption coefficient of card was deter- 
mined by experiment to be 2°28: card may be taken as 
cellulose, CgH,)V5; and the figures ‘given by Thomson, ‘ Con- 
duction of Electricity through Gases,’ p. 307, may be used 
to show that the coefficient of air must be greater than that 
of cellulose in the proportion of 8 to 7. In this last caleu- 
lation the absorbing power of H is neglected, which possibly 
makes the ratio too large ; but there are no data from which 
to determine the error ; it must be small, This gives k’=2°61. 
Lastly D=3°6 x 0012 =:00432. 


Hence eels kf 
IDk = ae 


aay : 2°61 
=79°7 x 3550 x 00432 x —— 


43°2 
= (ares 


whereas the ionization actually found, when the card was 
next the chamber and the nine silver foils on the outside of 
the card, was 70°3 as already stated. In this case therefore 
the ionization was somewhat overaccounted for. 

Generally the other experiments gave results of much the 
same kind; it would not be justifiable to expect more accurate 
confirmation under present conditions. 

The ordinary primary ray which was used in these expe- 
riments might well be replaced by one of the streams of 
homogeneous X rays which Barkla has shown us how to 
obtain from various metals. Recent papers by Beatty* and 
by Sadler t actually give results from which the desired 
information may be obtained in part, but neither author has 
had occasion to measure the value of X. Moreover there is 
no published determination of k’, the absorption coefficient 
of homogeneous X rays by air. Mr. Sadler has been good 
enough to tell me that he finds s’=93 for copper rays. 
Using this value, and taking A in silver to be the same as A 
in air, though it is probably greater, I find that on Beatty’s 
results about two-thirds of the ionization can be ascribed to 
cathode rays: the figures of the latter author give a rather 
smaller proportion. The agreement would be better if a 
larger value were assumed for X. Moreover these rays are 


* Camb. Phil. Soc. Proc. vol. xv. pt. v. 
+ Phil. Mag, March 1910, 


the Corpuscular Hypothesis of the y and X Rays. 405 


peculiarly liable to spend only a part of their energy in pro- 
ducing cathode radiation in the metal through which they 
enter ; some of the energy is spent on secondary X radiation ; 
or, which comes to the same thing effectively, some of the 
cathode radiation is liable to be reconverted into X radiation. 
In this way the measurement of Ik/\ becomes too small. 

There is another method by which it is sometimes sought 
to separate the ionization effect due to secondary 8 rays 
from the supposed effect due to the direct action of the 
y rays upon the gas, viz. the method of the magnetic field. 
Kleeman *, for example, has tried in this way to deflect from 
the ionization-chamber all secondary 8 rays, and has been 
able to reduce the ionization current to less than half its 
original value. Finding, however, that a considerable effect 
remained which he was unable to remove with the strongest 
magnetic fields at his disposal, he has concluded that this 
must be due to the direct action of the y rays upon the gas. 

The effect of a magnetic field is, however, a very difficult 
question to solve. It is to be remembered that the field may 
actually increase a @-ray effect in some ways while it lessens 
it in others. A §-ray path in the chamber may be lengthened 
by its being forced into a circular form, and the ionization 
due to the particle be made greater. Moreover, 8 particles 
are scattered by impact on the atoms of the surfaces upon 
which the magnetic field deflects them, and by successive 
impacts may travel considerable distances in spite of the 
field: for the field does no more than convert the rectilinear 
portions of the path into circular portions ; it has no influence 
on the direction which the particle will take after an impact. 
It cannot be asserted that the results obtained by the mag- 
netic deflexion method are yet capable of clear interpretation : 
further work in this direction is much wanted. 

Crowther has described an experiment from which he has 
drawn the conclusion that X rays passing through a gas 
lonize it directiy, and that consequently the cathode rays 
made by the X rays in the gas have no appreciable ionizing 
effect. He passed a fine pencil of X rays between two 
parallel plates so as to touch neither of them, and measured 
the ionization for various pressures of the gas. He found it 
to be very nearly proportional to the pressure : if cathode 
rays from the atoms of the gas were responsible for some of 
the ionization, the ionization due to them ought to show a 
marked decline as soon as the pressure of the gas is low 
enough to permit them to strike either of the plates, and so 
to leave their paths in the gas unfinished. He could not find 


* Proc. Roy. Soc. Ixxxii. 1909, p. 358. 


406 Prof. W. H. Bragg on the Consequences of 


a deficiency from the proportionality to pressure, as already 
said, and bence his conclusion. 

But it is clear that this experiment proves too much. One 
of the gases he used was methyl iodide. If X rays strike 
iodine atoms there is a very large conversion into cathode 
rays, as may be shown easily by scattering a little iodoform 
on a card through which X rays are entering an ionization- 
chamber when the current may be doubled under quite usual 
conditions. When the thinnest sheet of tissue-paper, equi- 
valent to 1°5 em. of air, is laid over the iodoform, this extra 
radiation is absorbed and the current returns to its former 
value. It isa clearly established principle that the effect of 
an atom upon an X ray is the same, no matter whether the 
atom is part of a solid or of a gas. Consequently there is a 
large production of cathode rays in the vapour of methyl 
iodide through which X rays are passing”, and a considerable 
fraction of the ionization of the gas is caused by these cathode 
rays. The amount can be calculated on the principles laid 
down above; but even if the complete accuracy of such a 
calculation be denied, it is still clear that the cathode ray 
ionization is large. Yet Crowther found there was none 
atall. Again, Mr. Edmonds has shown in this laboratory that 
if a hole is made in one of the parallel plates of Crowther’s 
experiment and a piece of wire gauze placed over it, cathode 
rays pass through the hole from the X-ray stream in quantities 
which show a large increase as soon as the pressure of the 
air is sufficiently lowered. The distance from the X-ray 
stream to the window is abouta centimetre, and the ionization 
current which is measured on the side of the gauze away 
from the stream increases rapidly relatively to the ionization 
in the air through which the X rays are passing: at first 
there is even an absolute increase in spite of the lowering 
of the pressure. The relation of the increase to the pressure 
alterations is Just such as would be expected if the ionization 
outside the gauze window was due to cathode rays made in 
the X-ray stream and passing through the meshes in the 
gauze. | 

If the ionization of the gas in an ionization-chamber across 
which y rays are passing is caused wholly by the 8 rays 
coming out of the walls of the vessel or out of the atoms of 
the gas, then, since the former of these sources of 8 rays 
is usually far more important than the latter, the ionization is 


* It is worth observing that in a mixture of methyl iodide and any gas 
of small atomic weight the iodine atoms would be responsible for a large 
ionization, but only a fraction of the ions would be formed from the methyl 
iodide molecules. 


the Corpuscular Hypothesis of the y and X Rays. 407 


due to an agent which does not change when the gas is 
changed, viz. the 8 rays from the vessel walls. The relative 


jonizations in different gases due to the y rays must be the 


same as the relative ionizations due to 8 rays ; and this is 


Fig. 3. 


GLE 


K= Cathode rays 
entering side 
Aamber 


found to be the case very exactly, unless there is such a mass 
of gas in the chamber that the second source of 8 rays becomes 
important. This occurs when the gas contains heavy atoms 
like those of iodine. The “atomic ionizations” by 8 and 
by y rays are set out below and show the close parallelism. 
They are taken from a paper by Kleeman *. 


de C. NM | 8 8. Cl. Br. | Ti; 
B ceccccees 0:18: 
5 gee PEE 018 


0°46 | 0-475 058 | 1:60 | 1-44 | 2°67 | 4°10 
0-46 | 0:45 | 0°58 | 160 | 144 | 2:81 | 450 


If any part of the ionization in the gas were due to a direct 
action of the y rays, and we were to reject the simple expla- 
nation just given, we should certainly find it extraordinarily 
difficult to explain the almost exact similarity of these two 
rows of figures. This would be the case on the entity hypo- 
thesis : if the y rays were supposed to be spreading pulses, 


* Proc. Roy. Soc. lxxix. p. 220, Feb. 1907. 


408 Prof. W. H. Bragg on the Consequences of 


differing therefore in every imaginable way from @ rays, an 
explanation would surely be hopeless. 

Considering all this evidence for and against the existence 
of a direct ionization of a gas by X and y rays, I would 
conclude that the entity hypothesis leads us to expect that 
there is no such effect, that many experiments fall in readily 
with this view, and that others are quite likely to show a 
like agreement when obvious defects have been removed. 

Let us therefore accept this simplification, provisionally at 
least ; and let us go on to consider a second problem of the 
ionization-chamber which may then be taken in hand with 
some success: the problem of the chamber of any form made 
wholly of any one substance. 

Suppose a block of any material to be crossed by a stream 
of y rays, and let us try to estimate so far as we can the 
whole length of track covered by @ rays in any element of 
volume in a second, irrespective of direction. The number 
will in the first place depend on the strength of the y radiation 
in the neighbourhood of that element of volume: after 
allowing for that, it will depend on two things only, (a) the 
number of 8 rays originated in eaeh unit weight of the 
substance, 2. e. the absorption coefficient of the y rays by 
the substance, (b) the weight of material traversed by each 
8 ray before it disappears. If different @ particles traverse 
different amounts of material, the average is to be taken: 
we may call such average the average range, or briefly the 
range. The important thing to observe is that the range 
need not be all in one straight line: the @ particle may 
make any number of twistings and turnings during its total 
path, and the range is the length of the path if it were 
straightened out, or rather the weight of material which the 
particle traverses. The deflexion oval and the scattering 
which the oval represents do not come into consideration 
at all. Let us say that & is the absorption coefficient of the 
y rays and d the range, then the sum of the tracks of B rays in 
a unit volume is directly proportional to Ikd, I being the 
intensity of they rays. It may be of some service to give an 
analogy. Ifk points were taken at random in each square 
centimetre of a sheet of paper, anda line of length d were 
drawn from each point, then the quantity of ink used and 
the quantity of ink on each square centimetre would be just 
the same, on the average, whether the lines were straight or 
curved or made up of any number of short pieces so as to be 
zigzag in form. The ordinary coefficient of absorption of 
8 rays is a compound of d, and of the dimensions of the 
deflexion oval. We are here dealing with a much simpler 


the Corpuscular Hypothesis of the y and X Rays. 409 


thing. If we take different substances and take I to be always 
the same, the ‘‘ 8 ray density ” in each substance is repre- 
sented relatively by kd. 

Suppose a cavity to be made in the substance. This makes 
no difference whatever in the value of kd anywhere in the 
metal, even on the borders of the cavity. This follows from 
the fact that every B particle has to cross a weight d of the 
substance: crossing the cavity does not count in its total 
path. The only inaccuracy in this statement arises from the 
fact that the value of I may not be the same in all parts of 
the substance that border on the cavity. It will be found to 
have little importance so far as our present purpose is con- 
cerned, and we will not take it into account. Then we can 
say that just as many @ rays cross each unit volume of the 
cavity as would cross it if it were filled with substance of 
the kind considered, or of any other substance having the 
same kd. The shape of the cavity is immaterial. We may 
in fact take it to be the inside space of an ionization vessel, 
provided only that the walls are thick enough to prevent the 
passage of 8 rays either way. 

It is curious but not uninstructive to consider that if we 
had a substance with no &, but with the power of reflecting 
every @ ray that fell upon it, and made a closed vessel of the 
substance, and shot ¥ rays across it, we should then get the 
following results. If a vacuum existed in the vessel, kd 
would be zero: if a single atom of any ordinary substance 
were placed in the vessel, kd would in time mount to its full 
value for that substance, and would not increase if the one 
atom were added to by putting in any number of like kind. 
J£ atoms of other kinds were inserted, there would be a 
compromise, the density of 8 rays becoming =k/2(1/d). 

To go back to the cavity in the substance traversed by 
y rays, the introduction of air into it makes little difference 
in the value of kd in different parts of it unless the kd of the 
substance differs considerably from the fd of air, and there 
is so much air that an appreciable fraction of §-ray energy 
is used up when a stream of such rays tries to cross the 
cavity. Hence the cavity must not be so big, nor the pressure 
of air inside it so great, that this source of inaccuracy becomes 
serious. If there were any doubt about it in a given case, it 
could be tested by varying the pressure of the air; if the 
relation between pressure and ionization required a curved 
line to represent it, it would be necessary to use the initial 
portion of the curve for which the pressure is small. This 
precaution is usually unnecessary, and we may take the 
ionization in the air of the cavity as proportional to kd. If, 


410 Prof. W. H. Bragg on the Consequences of 


therefore, we make a number of ionization vessels of different 
materials but the same form, and cause y rays to pass into 
them, the amount of ionization produced inside becomes a 
measure of the kd of that substance. The experiment may 
conveniently be carried out by making a thick lead ionization- 
chamber and inserting different linings. The y rays must 
of course be kept at the same strength inside each lining, or 
if not any differences must be allowed for. 

Mr. H. L. Porter has recently carried out some experi- 
ments for me in this way, the results of which are shown in 
the table below. The first column gives the material of the 
lining, and the second its thickness, which was enough to 
give the true value of kd in all cases except perhaps those of 
aluminium and cardboard. The third column gives the 
results obtained when the y rays had to pass through little 
more than the lead wall of the i1onization-chamber, which was 
0°47 em. thick, and the fourth the results when the rays had 
to pass through a screen of lead 1*1 cm. thick in addition. 
The figures are corrected for differences in volume and for 
differences in the strength of the y rays due to absorption in 
the linings. 


1, LL. ITI. IV. 
‘ Metal. Thickness | Ra unsereened.| Ra screened. 
of screen. | 
eed 122005 een eee ae 100 100 
RTRs So ohn, alate COG 16 58 68 
ACTA EME ily PD “21 | 47 59 
Bian itis. os OO 155) 4) 45 54 
Aluminium ........ Al 40 49 
ROME oy. sok ee "24 39 46 


The height of the chamber, which was cylindrical in form, 
was 15 cm.,and the diameter 9 cm, ‘The radium was plaecd 
on the axis of the cylinder, 10 cm. away from one end. 

The differences between the figures in the last two columns 
are really due to a change in the relative value of lead only. 
The rays have been so hardened by passing through the extra 
em. of lead that the absorption coefficient of the lead lining 
has fallen to the same value as that of the other metals. In 
the first case there is a special production of softer 8 rays by 
the lead which does not take place in the second. 

In these experiments the strength of the y rays:is not the 
same all over the cavity as it ought to be; but the inequality 


the Corpuscular Hypothesis of the y and X Rays. 411 


cannot have much influence on the relative values of kd for 
the different linings. Mr. Porter finds that the figures are 
indeed somewhat altered when the radium is moved about 
into different positions, but the alterations are such as would 
be expected from the variations in the quality of the y rays. 
In some positions the y rays pass more obliquely through the 
walls, and therefore through a greater thickness of lead, so 
that they are so much the more hardened. 

When all allowances for error are made we still have a 
set. of figures which show with considerable accuracy the 
relative values of kd in certain substances, and, since & is 
practically the same for all of them, the relative values of d, 
the range of the @ particle. It may be well to point out 
once more that this range does not give directly the power 
of penetrating screens of different metals; and indeed it 
varies in the opposite direction. The power of penetration 


_ depends also on the form of the deflexion oval which represents 


the scattering effect. In the definition of the range, and in 
the experiment which measures it, scatterings or reflexions, 
or so-called secondary radiations, have no part at all. In 
fact these experiments allow us to investigate separately one 
of the three main subjects of measurement already referred 
to, viz. the expenditure of energy along the track of the 
8 particle, since this must determine the length of the 
track. 

In order to complete a proper set of investigations of the 
8 particle phenomena, it is further necessary to find the form 
of the deflexion oval in all cases. This may be done by 
observing the scattering of the @ rays in various directions 
as they pass through very thin plates, since in such cases the 
scatterings are due to one encounter with an atom in each 
ease, as Madsen has shown (loc. cit.). The third subject of 
measurement is the conversion of form: so far as we know 
this is unimportant in the case of the @ ray, but it is just 
possible that an effect of this kind has been overlooked. 

Until satisfactory investigations have been made under 
these heads, it is impossible to find true foundations for cal- 
culation of the effects to be observed when sheets of material 
are placed over a substance emitting @ rays, that is to say, of 
the so-called absorption coefficients. For these coefficients 
must necessarily vary in a complicated manner from material 
to material and thickness to thickness, since they are involved 
functions of the range and of the scattering. It is too much 
to attempt a theory of the absorption of @ rays until these 
intermediate steps have been hewn into shape.. H. W. 


Schmidt has tried to fill up the gap by arguing back from a 


412 Prof. W. H. Bragg on the Consequences of 


large number of measurements of absorption and of scattering 
coefficients ft. He has defined two constants which he has 
called the “reflexion” and the “true absorption ” coefficienst. 
The former really represents roughly the facts of the deflexion 
oval, the oval being reduced to its axis, and the atom placed 
at various positions upon it; the latter represents the expen- 
diture of energy along the path. His two constants actually 
stand approximately for the two independent subjects of 
measurement which we have seen to be important in the 
case of the 8 ray. Itis therefore very interesting to com- 
pare his calculated values of the true absorption coefficient 
with the quantity d, which should be approximately in the 
inverse ratio. ‘lo what extent this is due is shown in the 
following table. The second column gives the values which 
Schmidt? calculated for the true absorption coefficients of the 
8 rays of uranium, 7. e. the values of his a/D. I do not 
think the values for radium are available. But it must be 
quite allowable to use the former instead of the latter, since 
the 8 rays of radium do not differ much in penetrating 
power from the 8 rays of uranium; while the values of «/D 
for uranium and for actinium are very much the same re- 
latively to one another, and yet the 8 rays of actinium are 
much less penetrating than those of uranium. The third 
column gives the relative values of kd, or practically of d, 
and the last the product of the figures in the two preceding 
columns. 


Substance. a/D. kd. a/D Xx kd. 
Mead Soe ish. Wee 1-69 100 169 
MRT Beets toner ea eee ee 2°14 (2°40) 68 145 (163) 
MAGIC: cn Se ccacemeeeeaee 3°00 59) 165 
GON: renee packer caer 3°08 54 166 
Adtaminiuin © 52.) cen. 3°26 49 160 
WARE \. 554 suse aeeees 3'32* 46 | 153 


* Calculated as for earbon from later figures given by Schinidt. 


The uniformity of the figures in the last column is only 
broken seriously by tin. Strange to say, the value 2°14 
which Schmidt gives for tin is quite out of line with the 
values he gives for all the other metals; if these values are 
plotted and a value for tin obtained from the curve we get 


+ See also McClelland and Hackett, Dublin Trans. 1907. ix. p. 37. 
{ Ann. d, Phys. xxiii. p. 671 (1907); Jahrb. d. Rad. 1908, p. 451. 


the Corpuscular Hypothesis of the y and X Rays. 413 


2°40, which leads to a value 163 in the last column. Schmidt's 
values of «#/D for the @ rays of actinium do not show this 
irregularity in the case of tin. 

The values ot d are clearly less for the smaller atomic 
weights. The whole track of a 8 particle in lead is actually 
greater, weight for weight, than in aluminium. Yet as is 
well known a 8 particle can penetrate a heavier screen of 
aluminium than of lead. The reason is that the lead atoms 
turn back the @ particles so much more than the lighter 
atoms do. In lead the particle finishes its course much more 
closely to iis starting-point; it is really a longer course, but 
there are many more turns in it. 

It is easy to see that there will consequently be considerable 
differences in the “ absorption curves ” of different materials; 
2. e. the curves which show the relation between the thickness 
of a screen placed normally to the path of a stream of @ rays, 
and the ionization in a chamber on the other side, a chamber 
which the rays can usually cross. A @ ray going through 
aluminium behaves rather more like an @ particle than when 
it yoes through lead, since it is less liable to deflexion in the 
former case, and the a particle has very few departures from 
a straight line course. The absorption curve of the 8 ray in 
aluminium should, therefore, be more like that of an a particle 
than the curve of 8 ray in lead. Now the a particle actually 
causes more ionization when screens are placed in its path, 
unless the screen is too thick, than when it is unimpeded ; 


Fig. 4. 


Weioht of absorbing material 


that is to say, the curve which is plotted with thickness of 
screen as abscissa and ionization on the other side of the 
screen as ordinate rises at first; subsequently it falls rapidly 


414 Prof. W. H. Bragg on the Consequences of 


to the axis of x. Experiment shows that the absorption curves 
for 6 rays in aluminium screens do really possess a trace of 
this peculiarity, for they fall slowly at first and much more 
quickly afterwards. On the other hand, the absorption curve 
for lead is more like an exponential curve, which is to 
be expected since scattering is the most prominent cause of 
absorption. 

Sir J. J. Thomson has recently published (Proc. Camb. 
Phil. Soc. xv. part v. p. 465) a theory of the “ scattering of 
rapidly moving electrified particles.” It seems to me to be 
inapplicable to the actual case because it considers scattering 
to be due to a multitude of small deflexions experienced by 
the particle in passing by the various centres of positive and 
negative force in the atoms, all the centres acting indepen- 
dently of each other. Apart from the question as to whether 
it is likely that the positives and negatives do not interfere 
with each other’s actions, the argument is limited to cases 
where the total deflexion is so small that the particle has 
hardly moved from its original direction when it emerges on 
the other side of the screen. ‘This is necessary because the 
deflexion is taken to be the average of a number of deflexions, 
and the reasoning tacitly assumes that all these deflexions 
are grouped symmetrically about the original direetion of 
the particle throughout the whole of the transit of the particle 
across the absorbing layer. The scattering of a pencil of 
8 rays is looked upon as a gradual opening out of the whole 
pencil, and the calculations refer to a state in which the 
absorbing layer is so thin that only slightly scattered rays 
are worth considering. Actually there is no such state ; 
however thin the plate the highly scattered rays are in a 
certain proportion to the slightly scattered rays, which does 
not alter as the thickness of the layer is increased, unless the 
thickening is carried too far. From the very first large 
deflexions must be considered. The many slight deflexions 
which the § particle experiences along the comparatively 
straight portions of its track are of no real consequence ; 
little more than in the case of the @ particle. Moreover, 
while the plate is still fairly thin, another important effect 
comes in, viz. the loss of speed; and it is by the mutual 
interplay of these two that the differences in the absorption 
curves are caused. Crowther (Camb. Phil. Soc. Proc. xv. 5, 
p. 442) shows absorption curves of aluminium and of platinum. 
The curves show the special characteristics just discussed ; 
but I think it is only by accident that the aluminium curve 
fits the formula derived by Sir J. J. Thomson. The curve 
for platinum will not fit the theory in the same way, and 


the Corpuscular Hypothesis of they and X Rays. 415 


Crowther supposes that secondary radiation must be present 
and be responsible for the want of agreement; but there 
does not seem to be any good reason for selecting secondary 
radiation as the cause of the error. On the other hand, the 
entity hypothesis leads naturally to a simple explanation of 
the general form of the curves both of aluminium and of 
platinum. 

In the case of @ and cathode rays there is very little 
accurate knowledge of the third of the phenomena which I 
have tried to distinguish above, viz. the conversion of form. 
The conversion of B rays into y rays is often doubted 
altogether; but it can hardly be safe to deny it, for if the 
number of y rays produced by a given number of 8 rays 
were relatively as few as the X rays produced by a stream 
of cathode rays, the effect produced by the y rays would be 
almost imperceptible. The conversion of cathode rays into 
X rays is, however, a very obvious and common process, and 
it is rather striking that so little work has been done to 
discover the laws of it. It would be a great help to know 
whether there is a critical speed or more than one critical 
speed at which an electron should strike an atom in order to 
get an X ray effect. Let us suppose that there is a speed 
which it is necessary for a cathode ray falling on a given 
atom to possess in order that the conversion may take place, 
which does not seem at all unlikely considering the general 
behaviour of X ray tubes. Let us suppose, further, that the 
critical speed increases with the atomic weight, for which 
also there is something already to be said. Then we seem 
to have a reasonable chance of explaining the very remarkable 
phenomena of the homogeneous secondary X radiations which 
Barkla has discovered. The explanation given by Barkla 
himself is not at all in accord with the arguments which I 
have tried to state above. He supposes the primary pulse to 
shake an atom in passing and make it give out its own cha- 
racteristic quivers. Bet this suggests that a single primary 
X ray is the cause of many secondary X rays. 

We have to explain why one single primary entity—an 
X ray—is replaced by one secondary X ray entity after 
collision with a certain atom, the energy of the secondary 
being characteristic of the atom not of the primary, and its 
direction of motion being also independent of the primary, 
i. e. of the direction of motion of the primary. We have to 
explain further why the X ray emitted by zine can excite 
the copper atom to emit its own characteristic X ray, and 
why the reverse does not take place, the copper X ray is not 
able to excite the zinc X ray. Let X rays from zine, that is 


416 The Corpuscular Hypothesis of the y and X Rays. 


to say secondary X rays coming off a plate of zinc on which 
sufficiently hard X rays are falling, be made to strike a plate 
of copper. Their energy is gradually converted into that of 
cathode rays, which possess a certain definite power of pene- 
tration, 2. e. a certain definite speed (or perhaps average 
speed) as Sadler has shown. These cathode rays possess 
more than the critical speed for copper; we may imagine 
them to scatter in the zine, losing all sense of original direc- 
tion very quickly and falling in speed. When they reach 
the critical speed for copper and the maximum conversion of 
form takes place, the characteristic X rays of copper will 
flash out in all directions. If they pass this speed without 
conversion their energy is spent merely on the copper atoms, 
transforming itself in the usual ways into heat. But if X 
rays produced by some means in a copper plate are allowed 
to fall on a zine plate, and there form cathode rays, the 
speed of these latter rays is below the critical speed for zine, 
and no X rays characteristic of zinc are produced. Thus all 
Barkla’s effects are qualitatively explained. Until the con- 
version of cathode ray energy into X ray energy has been 
more fully investigated, such an hypothesis can be no more 
than a provisional one, but it seems simple and reasonable, 
and suggests promising lines of research. 


In the foregoing pages I have tried to follow out the con- 
sequences of ‘adopting the “entity hypothesis” of X and 
y rays, and to show how we are led to modify our views of 
well known theories and our interpretations of well known 


experiments. Since there is so much to consider, the dis- 


cussion bas, I fear, been rather lengthy; but I think the 
result is simple. We are to think of each entity as possessing 
initially a certain store of energy which it spends gradually 
as it goes along, the result being ionization of the material 
through which it passes; there are no sudden accessions or 
withdrawals of energy; the path is not necessarily straight, 
but made up of a number of small pieces more or less straight, 
the deflexions or turnings being the results of intra-atomic 
collisions ; the 8 rays are very liable to such deflexions, and 
the cathode rays even more so. Certain conversions of form 
may take place, y into 8, X into cathode ray, and so on; but 
in such cases the energy is handed on, and in some cases at 


least the momentum. The essence of it all is the recognition 


of the individuality of each entity which is to be followed by 
itself from its origin through all its changes of direction and 
sometimes its changes of form, until its gradually diminishing 
energy becomes too small to render it distinguishable. 


XL. On Hysteresis Loops and Lissajous’ Figures, and on 
the Energy wasted in a Hysteresis Loop. By Professor 
Sitvanus P. Tuompson, D.Sc., P.R.S. * 


[Plate VI.] 
§ 1. gpa many physicists have attempted to find 


an explanation of the forms of the looped curves 
which express the hysteresis exhibited by iron and steel when 
subjected to cycles of magnetization. Physical explanations 
to account for their general shape have indeed heen given by 
Ewing and by Hopkinson. Neither of these pioneers, how- 
ever, offered any mathematical equations to express their 
forms ; nor, so far as appears, has any other person yet found 
any, though M. Pierre Weiss has put forward an electronic 
theory to account for the principal features. 

According to Ewing’s molecular hypothesis of magnetism, 
the act of magnetization consists in the orientation into a 
common direction of the axes of the elementary magnets 
constituted by the iron molecules which, when the mass of 
iron is in the unmagnetized state, are miscellaneous in their 
directions, the molecules being then arranged in groups 
within which the individuals are so oriented as to satisfy 
amongst themselves their various polarities in a more or less 
stable equilibrium. When a small magnetizing force is 
applied and gradually increased, the individual elementary 
magnets are at first merely slightly deflected towards the 
line of the magnetizing force, but still remain in their various 
groups. With larger magnetizing forces and increased 
deflexions of individual elements, the groupings, or some of 
them, become unstable, and break up as instability is reached ; 
the elements of the group then suddenly swinging round 
into a new configuration more nearly in alignment with the 
impressed magnetic force. The less stable groups will be 
first affected, the more stable afterwards, and the most stable 
will be the last to swing into alignment. Whenallor nearly 
all the groups have thus been broken up, any further increase 
in the magnetizing forces can produce but little effect, though 
an infinite magnetizing force might be needed to produce 
absolute alignment of every element. To deduce from this 
hypothesis an expression for the ascending curve of magneti- 
zation, it might be possible to apply the statistical method, 
under the assumption that the number and variety of the 
groupings is enormously great. The ratio d3s/d® would 
represent at each stage the differential permeability of the 

* Communicated by the Physical Society: read July 8, 1910. 


Phat. Mag.S. 6. Vol. 20. No. 117. Sept.1910. 9 2 F 


418 Dr. Silvanus Thompson on 


specimen, or the rate (per unit of magnetizing force) at 
which the magnetization was proceeding ; and this would at 
every stage be proportional to the probable number of groups 
that were coming into alignment, and to the magnitude of 
the aligning force. The expression would thus assume the 
form 


gO a.e-D-2) ; 

d® f 
where a and 6 are constants, the one denoting the maximum 
value of @38/di®2, the other the particular value of ® at which 
that maximum is reached. The difficulty of integrating this 
expression is not the only objection to it; for it would at 
best give only the ascending curve of magnetization, and 
additional assumptions would need to be made before it could 
be adapted to express the descending branch. 


§ 2. Hysteresis loops, as found by experiment, can how- 
ever be considered from a wholly different standpoint. 
Whatever the law connecting 3% and ®, the area enclosed by 
the loop measures the amount of energy lost in the iron in 
the cycle of magnetic operations ; the value, in ergs per 


cubie centimetre, being Ae 33 ..d®. Let it be assumed that 


the variations of # are such that % passes from the value 
+438, to —%, and back, through a regular cycle of values 
according to the expression 


5 = 36, cos 0; 


then, in general, the variations of ®, though they go through 
a cycle, will not be capable of being expressed by any such 
simple form, otherwise the shape of the hysteresis loop would 
be simply an ellipse, or, in the limiting case, a straight line. 
The fact that the magnetizing current (and therefore the 
magnetizing force) in choking coils and transformers does 
not follow a simple sine or cosine function is well known to 
all electrical engineers. In these instruments, if worked 
from a source of alternating electromotive force in which the 
wave-form of the supply is a pure sine-curve, the wave-form 
of the flux in the core, and therefore of the flux-density 3 in 
the core, will also be a pure sine-curve having a lag of exactly 
90° with respect to the impressed voltage curve. Or if the 
impressed voltage is expressed by the equation 


ye Vo sin 6, 
the flux-density will be given by the equation 


3 = — B, cos 0. 


— 


Hysteresis Loops and Lissajous’ Figures. 419 


But the current will not be capable of being expressed in 
any such simple fashion *. It bas also long been known 
how these curves of magnetizing current are related to the 
hysteresis loop. If the form of the hysteresis loop is known, 
then the form of the magnetizing current can be deduced 
graphically. An example will be found in Kapp’s ‘ Trans- 
formers’ (1908), fig. 56, p. 106. Assuming that the flux- 
density follows the equation given above, let it be plotted as 
a curve, with maximum ordinate equal to that of the hys- 
teresis loop. Consider any point on this curve, or the point 
on the loop having the same ordinate ; then the ordinate at 
the same instant on the current curve will be (on some scale) 
proportionate to the abscissa of the same point of the 
hysteresis loop ; whence it fellows that the current curve can 
be constructed, point by point, through the entire cycle. 
This process is followed here, save that for convenience the 
hysteresis loop is turned over through a right angle, so that 
the values of 3% are taken as abscissee, and those of #® as 
ordinates. In fig. 1 the hysteresis loop ABCDEF 
selected, has been taken from Ewing’s classical memoir of 


1885. 
Fig. 1. 


' 
—— ae awe te 


AIT 
2 


Relation of Current Curve to Hysteresis. 


Assuming a closed magnetic circuit, and an impressed 
voltage of a pure sine form, marked V, the flux-density curve 
is a sine-curve lagging 90° behind the voltage-curve. Then 
the current curve is derived by taking any point P on the 


* For a good example of a current curve, see Fleming’s ‘ Alternate 
Current Transformer,’ fig. 186, vol. i. p. 543 (edition of 1896). These 
irregular current curves seem to have been first observed by Ryan; see 
‘Transactions of the American Institute of Electrical Engineers,’ vol. iii. 
Jan. 10, 1890. See also the ‘ Electrician,’ vol. xxiy. p. 239, and p. 263, 
January 1890, and vol. xxv. p. 312, = uly 25, 1890. . 

2H 2 


420 Dr. Silvanus Thompson on 


hysteresis loop, projecting it by first turning its abscissa 
through a quadrant about the centre O, then tracing along 
horizontally to the pot Q on the flux- density curve, where 
a vertical line QM is dropped. Then a horizontal line pro- 
jected from P cuts QM in R, giving thus the corresponding 
point on the current curve, the peak of which, corresponding 
to the cusp of the loop, occurs at the time oem the flux- 
density curve is a maximum, and when the voltage curve is 
at its zero. 

It will be seen that the wave-form of the current curve 
reflects, in a certain way, the form of the hysteresis loop. If 
the loop is sharply cusped, the wave-curve will have corre- 
sponding sharp peaks. In fact, the loop consists of the two 
halves of the wave-curve, folded back one upon another, but 
with the ordinates differently spaced, exactly as if the wave- 
curve had been wrapped around a cylinder* and projected 
upon a plane cutting the cylinder diametrically through the 
two peaks of the curve. 

Now this current curve can be subjected to harmonic 
analysis, and its harmonic constituents discovered. Hach 
constituent will be a pure sine-curve or cosine curve. If 
each such constituent be drawn, and then be projected back 
by reversal of the process by which the wave-curve was 
obtained, the several constituents will reappear as separate 
closed curves ; and by the summation of these the original 
hysteresis loop can be reconstituted. It thus appears “that 
any hysteresis loop can be analysed into an harmonic series of 
closed curves corresponding to the various terms in the analysis 
of the current wave. An examination of these constituents 
of the hysteresis loop is the principal object of this com- 
munication. 


§ 3. In this graphic process, which is equivalent to 
wrapping the periodic curve around a cylinder, the area of 
the projected curve is equal to the integral, over the whole 
period, of products obtained by multiplying each ordinate by 
the sine of the angle at which it stands in the wave-curve ; 
abscissee in that curve being reckoned as values of angles. 
(The origin of the cycle is taken where the curve has its ° 
negative peak.) In symbols this is equivalent to 


f sin 6. W(0) . dd. 


* As in the graphic method of harmonic analysis of Clifford described 
by Perry in Proc. Phys. Soc. vol. xiii. 


Hysteresis Loops and Lissajous’ Figures. 421 


Now w(@), the function which represents the current-curve, 
may by Fourier’s theorem be expanded into the series 


A, sin 0+ A, sin 30+ A; sin 56+...... A, sin né 
+ B, cos 0+ B; cos 30+ B; cos 50+...... B,, cos né. 


Odd terms only are present, as in all alternating current 
work; and there is no constant term, because the mean 
ordinate is already zero. 

The constituent terms in the area of the hysteresis loop 
correspond therefore to the integrated products of sin @ into 
the successive terms of the above series. 


§ 4. To investigate the form of the constituent elements of 
the loop, let us consider a simple harmonic motion v= X sin 8, 
where @ stands for 27/t, f being the fundamental frequency, 
and X the amplitude. This motion is to be compounded, at 
right angles, with another simple harmonic motion 


y= Y,, sin (n6+¢,); 
where Y, is the amplitude, g, a possible angle of phase- 
difference, and n any (odd) integer giving the order of the 
harmonic. We have then to find an expression for the curve 
of which # and y are the coordinates. For simplicity we 


deduce the expressions where n=1, n=3, and n=5, that is 
for the first, third, and fifth terms of the constituent elements. 


First Term (Fundamental) ; n=1. 
We have 


= = sit Oy! se eee |. C1) 
Mes 


amet (0+ ¢;). F . a is Sa ‘ s 5 (2) 


Multiplying both sides of (1) by cos 1, we have 


v 


xX 


cos g,=sin .cos gy. 

Also 

= = sin 0.cos ¢; + cos #.sin ¢. 
l 


Subtracting this equation from the preceding, we have 


008 $i —  =— Cos O.sin py. Ati ee ua 


422 Dr. Silvanus Thompson on 
Multiplying (1) by sin q,, we get 
ze 
x 
Squaring (3) and (4), and adding them gives us 
(=) = (+) —2 xy, do; = sin’? d, . (5) 
This is the equation to an ellipse, such as is represented by 


fig. 2 (Pl. VI.). According to the values given to ¢, there 
arise three principal cases. 


sin @), = sin 7.sin gy... 6 =) ¢) 


Case (i.). If 


then 
sing;=+1 and cosd,=0, 
and the equation beeomes 
“4 


y? 
a ae 
le Ge 


This is the equation to an ellipse set orthogonally with 
respect to the coordinate axes as in fig. 3 (Pl. V1.). 


Oise (ii.). ae 
Pi = 0, sin dy, = (), and cos 1 = Ls 


and the equation becomes 
2 g 
ae ere 
eh: GM tno CF 
whence 
Xi 
Uae se Xx a) 


which is the equation to a straight line into which the ellipse 
shrinks as in fig. 4 (Pl. VI.). But its length is limited by 
the prior expressions, since w and y cannot exceed X and Y, 
respectively. 


Case (iii.). TE 


bb =T, sin p,=0, COS f= —1, 


Hysteresis Loops and Lissajous’ Figures. 423 


and the equation becomes 


or 


which is a straight line sloping the reverse way as in fig. 5. 

For all other values of ¢ the ellipse takes some intermediate 
form. ‘The sine-component of the first term in the harmonic 
analysis of the current curve corresponds to the orthogonal 
ellipse ; the cosine-component to the oblique line form. 
All the intermediate forms of the ellipse could be obtained 
by wrapping a sine-curve of period T around a cylinder of 
diameter ‘l'/7 and projecting in appropriate directions upon 
planes parallel to the axis of the cylinder the apparent outline 
of the sine-curve. 


Tuirp Term (Third Harmonic) ; n=3. 


Here the two equations are 


op St aU AR OD AL) oe} et oe QL) 


¥, = sin (38+ $s), Uo 0 OES ea ae 0) 


= sin 3@.cos¢3;+cos30.sing;.. . (2a) 


But 
sin 36 = 3sin@—4 sin’ @, 


by known trigonometrical relations. 


Inserting for sin @ its value from (1), we get 
git, SO eee Pouca dah 4 
Substituting this value in (2 a), we deduce 
(<- ) cos $3— v = —cos3@.sing;.. . (4) 
Also multiplying (3) by sin d3, we have 


52 427Ve ; : 
(5 — $5) sin f.=sin 34 sin fy. ahs st (ek) 


424 _ Dr. Silvanus Thompson ox 


Squaring (4) and (5), and adding, we get 


a2 Aa\? WA Qa |) Ape 
(3-3) +(¥)- a(x - xe vy, . cos @3=sin? 3. (6) 


This is the equation to a figure having the general form of 
fm 6 (El. V1.); which is indeed the well-known Lissajous’ 
figure, compounded of two vibrations the frequencies of 
which are as 1:3. It could be obtained by taking three 
complete sine-waves, each of period T/3, and wrapping them 
around a cylinder of diameter T/7r. 

Again there arise three cases :— 


Case (i.). If d;=47 or 37, then 
sing@dz;=+1 and cos¢;=0, 


and then the equation becomes 


De Ae ) yy )= 
See a a ee Ife 
& xX? (¥. 


Here the figure is symmetrical with respect to the axes, 
as In fig. 7. “Tt i is, for the third term, what the orthogonal 
ellipse is for the first term. 


Case (ii.). If 6;=0, then 
sing;=0, cos d3;=1, 


and the equation reduces to 


Here the trilobate form has shrunk to the form of the 
curved line (fig. 8) precisely as the ellipse shrank to the 
oblique line of fig. 4. This line is subject to the limitations 
that # and y cannot exceed X and Y3, respectively. 


Case (iii.). If d3=7, then 
sing;=0, cos?;=—1, 
and the equation becomes 


Oty ila ae 
a ge 


the graph of which is fig. 9 


0 A i 
a ee eee 


Hysteresis Loops and Lissajous’ Figures. 425 
Firra Term (Fifth Harmonic); »=5. 


The two equations now are 


=m RIC eee sos MG date OO ORT) 


eS 


ae sie COE AY 1 AR OS Sa EP a 

= sin 50. cos ¢5+cos 50. sings. . . (2a) 
Sut 

sin 50=5 sin @— 20 sin? @+ 16 sin’ 0. 

Inserting for sin @ its value in (1), we get 
Fee 202 Lon? ; 
eee ee (): 
Substituting this value in (2 a), we deduce 


(Bz _ 20x? - Loi? 
X A KP 


sin 50 = 


) eos o3— - =—cos5@.sinds. . (4) 
5 


Also multiplying (3) by sin };, we have 


s — oe a es ) sin gd;=sin50@.sing;s. . (5) 


Squaring (4) and (5), and adding them, we get 


5a 202° pH 5a 2022 ee : 
Mx! xa t a) + (¢ ) (5° xa + a cos d;=sin* d;. 


This is the equation of the general Lissajous’ figure of the 
fifth order, representing the result of compounding two 
vibrations having relative frequencies of 1 : 5, and a general 
form like fig. 10 (Pl. VI.). 


As before, three cases arise :— 


Case (i.). If 63;=427r or 37, then 
sing@;=+1 and cos¢,=0, 


and then the equation becomes 


ba, 20H Ga Ne ia 
xo tyr) +) =4 


which is symmetrical as in fig. 11. 


426 Dr. Silvanus Thompson on 
Case (ii.). If ds;=0, then 
sings=0, cosd?s=1, 
and the equation reduces to 


De 20a? agar ae 
XX Oe aaa 
corresponding to fig. 12, subject to limitations as before. 
Case (iil.). If ¢d3=77, then 
sin@;=0, cos¢ds;=—1, 
and the equation is 


5a | (20a a GaP a aay 
Kc Ry My ate” 


which is the equation to fig. 13. 


§5. The Higher Terms. Generalized expression for the 
Lissajous’ curve of the nth term. 


The expression for any higher term has the general form 


2 X: ; 
G?+ co —2G, “y . cos &, =sin’ d,, 


and, in cases where ¢, =0, to 
a 
G+ y =0, 


in which expressions G, is written for the series 


n—1 
G, aS Ca Ope re, = a hea — 1 ; 


_@ 
n? 


where 
n(n? —1?)(n?— 3?) ...... (n?—7 —2”) 
(Ssh EN ap ile i Fi 2 IN eT AERA 


where r is an odd number greater than unity. 


C= 


§ 6. It is now possible to restate the proposition at the 
end of § 2 in the following way. <Any hysteresis loop can be 


‘As Ne 
Lie 


Hysteresis Loops and Lissajous’ Figures. 427 


analysed into an harmonic series of Lissajous’ figures of the kind 
considered in the $§ 4 and 5. 

A number of examples of hysteresis loops were chosen, 
and subjected to harmonic analysis, to ascertain what con- 
stituents were present. The loops chosen relate to various 
kinds of iron and steel, hard and soft, solid and laminated, 
taken by various methods ; a wide selection being made in 
order to ascertain the physical significance of the several 
constituent terms. 

In carrying out the analysis the author used the simple 
approximate method described by him to the Physical Society, 
Dec. 9, 1904, vol. xix. Proc. Phys. Soc. p. 443, based on an 
arithmetical process originated by Archibald Smith and 
generalized by Runge in the Zeitschrift fiir Mathematik und 
Physik, vol. xlviii. p. 443, 1903. It was found that for the 
present purpose it sufficed to ascertain the harmonic sine and 
cosine terms up to the eleventh, and therefore to employ 
twelve equidistant ordinates in the half-pericd. The work 
proceeded on the lines of the simple schedule given by 
the author on p. 448 of his former paper, with a slight 
modification to enable the origin of abscissee to be taken 
not at the point where the ordinate has zero value, but 
at that point where the ordinate has its negative maximum. 
At first the values of the twelve ordinates required for the 
analysis were taken from the current curve plotted, as ex- 
plained above in § 2, from the hysteresis loop. But it was 
seen that it was unnecessary to draw the current curve, and 
that the values of the ordinates might be taken direct from 
the hysteresis curve, by taking them not equidistant, but at 
places corresponding to equidistant points in the axis of 
abscisse of the wave-curve, which points, when the curve is 
wrapped round a cylinder, will no longer appear equidistant. 


§ 7. The following are the results :— 

Example I. fig. 14, Pi. VE., Ewing’s hysteresis loop for 
pianoforte steel wire, in state of normal temper, being fig. 11, 
pl. lviii. of Philosophical Transactions, 1885. 

The analysis of the values of ® gives the following co- 
efficients of the harmonies up to the eleventh order :— 


Sine Terms. Cosine Terms. 
Ap 5a ee B, = —45°4 
A; = Ck Bz; = —20°6 
ee 0-7 iBz=—10°8 
A, =— 07 Be = —.5°7 
Ay =— 0:05 B,-— 41 

. A= “= 0:06 by — 34 


428 Dr. Silvanus Thompson on 


It will be seen that the values of the sine terms beyond A; 
are negligible, and are not greater than the errors due to the 
approximate nature of the method. The cosine terms are all 
negative and of decreasing values for the successive orders. 
In the Plate the wave-curve has been given for comparison, 
and on it the components A,, A3, £,, Bs, and 5; have been 
plotted in dotted wave-curves. For comparison with the 
hysteresis loop its chief components have also been drawn in 
dotted lines: the ellipse corresponding to A, ; the trilobate 
curve of A,; the oblique straight line of 6,, and the curves 
for B, and B;. 

Hxample II. fig. 15, Pl. VI, Ewing’s loop for annealed 
iron wire, being fig. 5 of plate lviii. of Puil. Trans. 1885. 

The analysis gives :— 


A; =3°9S B, =—7'38 
A, =2'14 B, =—4:74 
A, =1°36 B. = — 3-04 
A, =0°88 B, =—3'78 
A, =" FG Bb, =—2'1]4 
Ay=0714 By = —1:90 


The ellipse and the straight line, corresponding respectively 
to A, and B,, have been added in dotted lines in the figure. 

Huvample ILI. fig. 16, Pl. VI., Ewing’s loop for annealed 
iron wire, being fig. 6, plate lviil. of the same memoir. 

The analysis gives :-- 


Ay, =4°2 B, =—30°4 
As = Bs = —25°5 
A ged Be = —A17°9 
Ay. =0°2 B,=— 67 
Ay. =Did By =— 495 
AG =O B= — 0°5 


Example IV. fig. 17, is taken from Lord Rayleigh’s 
paper in the Phil. Mag. xxiii. pp. 225-245, 1887, or 
Scientific Papers, 11. p. 593, and is the loop obtained with 
very small magnetizing forces on a specimen of “ rather 


hard Swedish iron.” 
The analysis gives :— 


Ay = 0553 B, =—1:022 
As =0°088 B; = — (0094 
A; =0-006 Bou 20-046 
A, =0-002 B, =—0°-025 
A, =0°005 By = —0:023 
A,,=0°000 = —(0°012 


——s <r 


Hysteresis Loops and Lissajous’ Figures. 429 
Example V. 


The next example is taken from a memoir of K. Angstrom 
in the ‘Proceedings’ of the Royal Swedish Academy of 
Sciences, 1899, p. 257, where the curves are given without 
any scale-values. They relate to a rod of steel containing 
0-2 per cent. of carbon. Fig. 18 was observed by a magneto- 
static method, fig. 19 by using an alternating current of 20 
periods per second, fig. 20 by using one of 60 periods per 
second. 

The analyses, on an arbitrary scale, are :— 


Fig. 18. 
| Aja Oat iB, = —71-29 
Ay ss) GE Pepe 2 (V9 
A; = —0°02 B; = —0°40 
A, = —0°005 = —0°23 
A, =—001 By, = —0-24 
Aged By=—0°24 
Fig. 19. 
Ae =a eel B, =—8'45 
Ag = US B, = —0°42 
Ae = ae B; =—0°04 
Ay 1 86 ea) "02 
Ay = 05 By =—0:07 
A,j= 004 By=—0:07 
Fig. 20 
A, a gt 2 B, = —8'57 
A. Ee ea (c( 
Apes WO B= 0-33 
A. = —0:09 B= . 0-28 
Ay =—0°04 Bh, = —001 
Ay,=\) 0°09 By, = —0 03 


The small scale of the original drawings of these three 
loops makes the values of the higher harmonics quite un- 
reliable. But the comparison is of interest as showing the 
effect of eddy-currents in the substance of the rod in widening 
the loops, and in increasing both A, and A3. . 


§ 8. Work done in the cycle.—It was early pointed out by 
Warburg and by Ewing that the work spent in carrying the 
iron through a cycle of magnetizing operations was repre- 
sented by the area of the hysteresis loop. We now consider 
this from another point of view. 

Whatever work is spent in magnetizing the iron is derived 
from the electric energy which is imparted by the circuit, 
and this, at any instant, is proportional to the product of the 


430 Dr. Silvanus Thompson on 


current and potential at that instant. If the voltage is re- 
presented by the expression 


V=YV, omg, 


where p stands for 2/f, and if the current, being some 
periodic function of the time, is represented as 


Nas 

C=(pt), 
then the element of work imparted to the circuit during time 
dt being CVdt, the work given to the iron (if the copper 
resistance 1s negligible) daring one cycle will be 


rT 
Vo sin pt .yr(pet) . dt. 
0 


But yr(pt) consists (see § 3) of a series of harmonic sine 
and cosine terms. ‘The quantities which will be formed by 
multiplying the members or that series by sin pt, and inte- 
grating each product over a whole period, will fall under 
three kinds, the values of which are known, viz.:— 


T 
(i.) i sin pt. A, sin npt . dt =0, (except when n=1); 
0 
er ‘T 7 . 3 
(11.) sin pt. B, cosnpt . dt=90, (in all eases); 
0 
Bs ia : fh 
{iii.) \, Agni SI Didi == A ; ; 3° 


That is, the only work done in the cycle is that done by that 
constituent of the current which is in phase with the voltage, 
namely, its fundamental sine-term. All other constituents 
are wattless. And since the area of the loop represents the 
work done, it follows that the area of the hysteresis loop is 
equal to the area of the orthogonal ellipse which is its funda- 
mental constituent of the sine series. The true and funda- 
mental form of every hysteresis loop is therefore an orthogonally 
placed ellipse. Ail departures from that form are wattless— 
are mere distortions which involve no expenditure of energy. 
The area of the hysteresis loop is proportional to the maximum 
value of % and to the amplitude of the first sine term into 
which the values of %# (corresponding to the values of the 
current) can be analysed. If the value of the amplitude of 
that sine-term be denoted by ®,, then the area of the loop is 

7 


accurately given by the expression in Bmax. X #1; being the 


FTysteresis Loops and Lissajous’ Figures. 431 


area of the fundamental elliptical constituent. To test this 
conclusion the areas of the three loops, figs. 18, 19, and 20 
were planimetered for comparison with the values of A: 


Planimeter reading. A. Ratio. 
Fig. Roose 33°8 0°57 594 
Ge 140: Drak 605 
Fig. Benak 167°5 2°04 610 


All the constituent curves belonging to the higher orders 
have zero areas; the lobes formed by the crossing of their 
outlines being alternately positive areas and negative areas. 
This is only another way of saying that the integrals (i.) and 
(ii.) above are always zero. As for those of form (ii.), they 
are obviously so, as all cosine constituents shrink up to mere 
lines. 


§ 9. Presence of Eddy-Currents. 


If the hysteresis loop has been produced by some slow 
process, absence of eddy-currents may be assumed. But 
this is by no means the case when alternating currents of 
ordinary frequencies are used, even if the iron be finely 
laminated. It therefore remains to be seen how the presence 
of eddy-currents will affect the size and form of the hysteresis 
lcop. The eddy-current, being a secondary current, will be 
of pure sinusoidal form only if the inducing electromotive 
force be of a pure sinusoidal form, and if the resistance 
and permeability be constant also. Butit is not necessarily 
in phase with the impressed electromotive force, but may lag 
by magnetic reaction ; and indeed, as is already known, lags 
by different amounts at different depths below the surface of 
the iron. Assuming equal permeability and resistance in the 
different layers, the effect of the eddy-current will be repre- 
sented with sufficient accuracy by a sine-curve lagging by 
an amount that will depend on conditions into which there 
is no need here to enter. For here, agai, the only effective 
component—etfective that is in the sense of involving ex- 
penditure of energy—is the sine-component in phase with 
the voltage ; and the element which the sine-component con- 
tributes to the loop is an orthogonal ellipse. So tar as it lags 
it possesses a cosine-component, and this contributes to the 
loop only an oblique line, shearing the loop over; but this 
constituent is wattless. Harmonic analysis cannot of itself 
distinguish as to how much of the fundamental elliptical 
constituent of the loop, or of the fundamental sine-component 


432 Dr. Silvanus Thompson on 


of the magnetizing current, is due to eddy-currents, and how 
much to hysteresis. It is, indeed, aiready known * that the 
effect of eddy-currents is to widen out the loop elliptically. 


§ 10. Lffect of the Higher Sine-Constituents. 


The presence of the third harmonic has been noted in § 7 
above. Indeed, it is usually present. A fine example is 
afforded by the curve, fig. 21, which is taken from F'leming’s 
‘Alternate Current Transformer,’ vol. ii. p. 486 (edition of 
1892), which affords the following analysis :— 


Fig. 21. Fig. 22. 


Ai eal es B, 
A, = 14g 1 Bs; 
A, =—)(074 B; 
A, = 4062 ae EHO 
A, =— 0°29 Bo = +) O22 
Ay=+ 0:03 By=-+ O°d7 


In the memoir of Angstrém already quoted, a curious 
eurve, fig. 22, nearly a pure third harmonic is given, having 
been found by him as a differential hysteresis loop repre- 
senting the difference between two steel rods, one having 
0-2 per cent. the other 0°8 per cent. of carbon. 

It is obvious that the effect of superposing a third-termn 
sine-constituent upon the fundamental ellipse will be to 
narrow it at the middle. It will widen it toward the ends, 
the widening beginning at a distance of 0°134 of the semi- 
major axis from the ends, as in fig. 23. 

If sine-constituents were present, of all orders, in the 
following proportions, i 

A,=1, A,=i, Amt ea F 

piece Angstrim, op. citat.; and Heinke, Die Electrephysik (1904), 

p. 500. 


Hysteresis Loops and Lissajous’ Figures. 433 


the form of the loop would become a pure rectangle, as 
fig. 24. 


The half-width of any static hysteresis loop, which is | 
commonly taken (following Hopkinson) as a measure of the 
coercive force, is independent of all the cosine terms, and is 
in all cases equal to Ay — A;+ A;—A;z+Ay— Ke. 

For loops of equal height (that is equal maxima of flux 
density) the coercive force is not proportional to the work 
spent in the cycle ; for the work spent in the cycle is repre- 
sented by the area of the loop, and as shown above, this is 
invariably proportional to .A;, while the coercive force is 
represented by the sum of the series named in the previous 
sentence. Since in static hysteresis loops the fifth and higher 
terms are practically absent, the coercive force is proportional 
to A,— As. 


§ 11. Effect of the Cosine-Constituents. 


As already seen, the fundamental cosine-constituent is an 
oblique line. It is in reality a double line, traversed upwards 
during half the cycle, and downwards during the other half. 
The effect of superposing this constituent upon the funda- 
mental ellipse is to shear it over. Cosine-components are 
always negative, corresponding to lagging constituents of the 
wave-curve. They shear the %6-# curve over toward the 
right, on the ascending side. Ifa mass of iron undergoing 
magnetization is traversed by an air-gap, or is constituted as 
a non-closed magnetic circuit, the reaction of the air-gap 
brings a lagging constituent into the magnetizing current, 
imposing a negative cosine-constituent upon the loop, and 
shearing it over. 

The presence of negative cosine-constituents in hysteresis 


Phil. Mag. 8. 6. Vol. 20. No. 117. Sept. 1910. 2G 


434 Dr. Silvanus Thompson on 


loops is specially marked in those beaked forms which are 
obtained when the magnetization has been pushed to high 
degrees of saturation: the diminished permeability of the 
material resulting in a diminished reactance, and therefore 
in a disproportionate increase in the magnetizing current. 
This is well seen on examination of the analysis of figs. 14, 
15, and 16. Cosine terms of the higher orders are responsible 
for the distortion of the ellipse into the characteristic two- 
beaked form. Fig. 25 shows the result of superposing a 
negative third cosine-constituent upon the ellipse. But this 
particular figure, resembling a capital §, could not result 
from any experiment, as neither the ascending nor the de- 
scending half is single-valued. No experimental curve* could, 
on ascending from the point where it crosses the # axis, 
curve backwards toward the % axis, and then recurve from 
that axis. Such a curve would be unstable; and the 
ascending branch could, at most, ascend parallel to the % 
axis before turning to the cusp. Is it too remote to speculate 
that the vertical portions of the loops experimentally found | 
for soft iron really do represent instabilities, and mask the 
true forms of the loops? 

The superposition of a negative fifth-order constituent is 
shown in fig. 26; but, again, the only possible cases must be 
such as to yield single-valued resultants. 


It will have been noticed that in the analyses of some of 
the loops the amplitudes of the negative cosine terms appear 
in a descending set of values. If these amplitudes are of 
relative magnitudes as 1:3:}:4 &c., the resultant curve 
will have its middle portion truly vertical. Fig. 27 gives a 


* See, for example, P. Holitscher (dnaug. Diss., Ziirich, 1900), 
“ Experimentelle Untersuchungen uber den remanenten Magnetismus 
des Hisens,” plate 6. 


Hysteresis Loops and Lissajous’ Figures. 435 


set of such curves up to the thirteenth order, together with 
their resultant. If this resultant is then superposed on a 
fundamental ellipse, the result is a characteristic two-beaked 


loop. 


Fig. 27. 
( 


yh f i \ 
/ t ' 1\ 
/ / x ! A 


The horizontal length from the cusp to the vertical axis 
(2. e. the amplitude of 8 at its maximum) is in all cases equal to 


B,+B;+B;+B,, &e. 


§ 12. Hifect of Harmonic Constituents of Even Orders. 


If constituents of even orders could exist, they would 
produce distortions of the loop, such as are shown in fig. 28. 
No such distortions have ever been observed. 


Fig. 28. 


CosInE COMPONENTS. 


Cea SS = Se 


ae 


\ 
i 
] 


436 Hysteresis Loops and Lissajous’ Figures. 


§ 13. Hysteresis is commonly regarded as an irreversible 
process, and as such involving a degradation of energy into 
heat. But in view of the present analysis of the hysteresis 
loop it is necessary to revise this opinion. In the first place, 
no energy is wasted in producing any cosine component of 
the loop. In the case of the first cosine term—the oblique 
line—the distortion is a mere shear. ‘True energy is spent in 
half the cycle in producing the magnetic flux, but that energy 
is returned to the magnetizing circuit during the other half 
of the cycle, exactly as it is in the case of the production of a 
magnetic flux in a solenoid devoid of iron. The like is true 
of the higher cosine-components. Also the sine-components 
higher than the first represent reversible processes. The 
only component which represents an irreversible process is 
the fundamental sine-component, the ellipse, itself due to that 
component of current which is in phase with the voltage. 
This irreversible part is due in detail to an energy-waste 
which at every instant is proportional to the square of the 
magnetizing current, and is in phase withit. True hysteresis 
waste accompanies the current, and does not either lag 
behind it nor precede it. True hysteresis does not cause any 
lag in the current, being necessarily simultaneous with it. 
True, the curve of the flux-density always lags 90° behind 
the voltage curve, and therefore 90° behind the effective 
component of the current curve. Of the reversible com- 
ponents, it is the first cosine term which is concerned in 
the lag of the current, and the lagging component of 
current is wattless. The higher cosine-components conjointly 
produce additional lags, and are also wattless. The higher 
sine-components, also wattless, do not cause any lag of the 
current asa whole. The name “hysteresis” was originally 
given * to the phenomenon to connote an effect which lagged 
behind its cause. The term is now usually restricted to the 
phenomenon of energy-waste. But the energy-waste does 
not involve or produce any phenomenon of lag. On the 
contrary, as it is a simultaneous phenomenon, its presence 
actually produces an advance in the phase of an otherwise 
lagging current. The greater the reluctance of the mag- 
netic circuit, the greater the angle of lag. The less the 
permeability which enters into that reluctance the greater 
is the angle of lag. The lagging components, that is the 
cosine-components of the loop, so far as they are not due to 
eddy-currents, depend solely upon the reduction of the per- 
meability of the iron during the process of the cycle of mag- 
netization. The lag is in fact due to components other than the 
fundamental component which represents the true hysteresis. 


* Ewing, Phil. Trans. 1885, pt. ii. p. 524. 


We cra 


XLI. On the Precise Effect of Radial Forces in opposing the 
Distortion of an Elastic Sphere. By J. Prescort, M.A., 
Lecturer in Mathematics, Manchester School of Technology *. 


en question is part of a more general one which is 

worked out very fully in Love’s ‘ Theory of Hlasticity ” 
(arts. 1706-178). Since, however, there is an important error 
in Loye’s solution, an error of principle and not of calcu- 
lation merely, I have presumed in this paper to point out the 
error and give the correct solution. It is all the more 
necessary to point out the error because, not being a very 
obvious one, it might pass a long time unnoticed in such an 
excellent book as Love’s. 

It is supposed that straining forces act on a solid homo- 
geneous sphere which have a potential of the form V+U, 
where V=/(7) and U is the sum of several spherical solid 
harmonics. ‘Then it is reasonable toassume, as in Love, that 
the radial displacement will be composed of solid harmonic 
terms also. 

Let us suppose that the matter which, in the unstrained 
sphere, was distributed over the sphere of radius 79, is now 
distributed over the surface whose equation is 


rT=T7, == Ip 


7, being constant for each shell, and o being a small quantity 
which is a function of 7; and angular co-ordinates. The 
equation (1) is thus the equation of a family of nearly 
spherical surfaces with 7, as parameter. 

Let us also suppose that the coordinates of a particle on the 
surtace whose parameter is 7 are &+u, yytv, 2, +w where 
#1, Y\;, 2, are the coordinates of a point on the sphere of 
radius 7. 

It will be seen from the above that we are considering the 
displacement of every particle to be composed of two distinct 
parts, namely (1) ong in which every shell which was sphe- 
rical in the unstrained state and had a radius 7, is strained 
into a shell of radius7,; and (2) displacements measured from 
the strained spherical surface. Now the radial forces with 
potential V will produce a radial strain, and the other type 
of strain can be produced by the harmonic forces. 

We will now suppose that the radial forces, if acting alone, 
would produce the radial displacement from 19 to 7,, which, 
we shall assume, is not sufficiently large to alter the density 
appreciably, or to affect the elastic properties of the sphere. 
If now we completely iguored the radial forces we could find 

* Communicated by the Author. 


438 Mr. J. Prescott on the Precise Effect of Radial 


the additional displacements due to the harmonic forces, and 
it might appear that these would be the displacements repre- 
sented by u, v, w above. But if the radial forces are very 
much larger than the harmonic forces, it is obvious that they 
have the effect of reducing the deviations from the mean 
sphere. When any spherical shell is distorted from the 
spherical form the particles of the shell are in places where 
the radial forces are unequal, and thus differential forces are 
brougnt into play which are of a similar type to those of the 
harmonic forces. 

The difficulty arises from the fact that x, y, and ¢ are used 
with double meanings. In the expressions for the potential of 
the external forces and for those forces themselves, they mean 
space-coordinates, Thus if V denotes the potential of the 
external forces, we suppose V expressible in terms of these 
space-coordinates thus 


V=P(e 952), 


and the forces are derived from this by differentiation and 


expressible in the form 


X= — ar =e — BAG, Wve), les 

Now there is a second meaning to w, y, and z. They are used 
as the distinguishing coordinates of a particle of the elastic 
body even after the body is strained. They are the space- 
coordinates of the position of the particle before strain. The 
space-coordinates after strain are represented by w+u, 2+, 
and z+w. It is clear, therefore, that the «-force at the point 
occupied by the particle whose distinguishing coordinates are 
(z, y, 2) 1s, after the strain, F,(@+u, y+v, z+w). Now 
these external forces at the point (v+u, y+v, <+w) are in 
equilibrium with the stresses at that point. But the stresses 
are regarded as functions of the distinguishing coordinates 
of the particles of the body; so that if P=d(w, y, z) is one 
of these stresses, we mean by ¢ (a, y, 2) the stress at that 
point of the body which was at (2, y, z) before strain, and 
which is at (v2v+u, y+v, 2+w) after strain. Thus in the 
equations of equilibrium we must express the stresses in 
terms of (a, y, <) and the external forces in terms of (4+, 
y+v, 2+w), and that is my method. 

In most problems the external forces in the displaced 
positions would differ so little from those in the original 
positions, that it would be superfluous to take account of the 
difference. But in the problem considered here the differ- 
ence in the case of the radial forces is of the same order of 


Forces in opposing Distortion of an Elastic Sphere. 439 


magnitude as the harmonic forces. Professor Love does 
not neglect this difference, but he allows for tt in a wrong way. 
Instead of modifying the bodily forces throughout, he assumes 
that the difference is properly accounted for by treating the 
bodily forces on the matter displaced outside the original 
bounding surface as if it were a surface-traction. This is a 
consequence of regarding the coordinates which occur in the 
equations of equilibrium as the space-coordinates of the 
particles of the strained body; whereas, as I have pointed 
out, they are the space-coordinates of the particles of the 
unstrained body. Surface-tractions are forces applied at the 
boundary of the body, and not forces applied at that surface 
which was the boundary in the unstrained state. 

When the harmonic forces do not act, the radial force acting 
on the spherical shell whose new radius is 7,, has a potential 
f(7;.) But when the spherical shell is strained to the surface 
whose equation is 


r="f+o 


the radial force acting at different points of this shell is 
derived from the potential 


V=/(r,+¢) 
=f (7) +of'(r,), nearly. 


The first term of V is the one that causes radial strain, and 
it has no effect whatever on the value of c. Since, in the 
rest of the paper, we shall only be concerned with the de- 
viation of the shells from the spherical form, we can ignore 
this term both in the differential equations and in the boundary 
conditions. 

But the second term in V causes displacements from the 
mean spheres ; and if its magnitude is of the same order as 
that of the harmonic forces, it must be added to the potential 
of these forces. 

We shall assume that 


F— Den Onis 


where eé, is a small coefficient and Q,41 1s a spherical solid 
harmonic of order n+-1. 

We shall write 7, «, y, 2 instead of 74, 2, y1, 2, to save 
labour in writing wherever no ambiguity can arise. 

In problems concerning the tidal action of the sun and 
moon on the earth and the action of ‘‘ centrifugal force ’’ due 
to the earth’s rotation f(r) is proportional to 7? ; and therefore 


440 Mr. J. Prescott on the Precise Effect of Radial 


we may put/(7,)=Kr,. Thus the term which has to be 
added to the potential of the non-radial force is 


WwW ra Kr d€nQn+ le 


We have now to find particular integrals of the equations 
of equilibrium dueto the term W, and to determine its con- 
tribution to the boundary conditions. 

The equations of equilibrium are three such equations as 


ow ! 
+m) se FN ET Pe =0, 10) Ree 


the symbols used having the same meaning as in Love’s 
book. 
We shall obtain particular integrals by assuming that 


08) ean OP 
"On mee tease 
Then equation (4) gives 
E W ” 
NEE wet +p Sa) ~ (5) 


whence 

At 2) Vo=—pW. «ss 7 ete 

Now if Wy denotes a solid harmonic of order (n+1) we 

know that 

V2 (7? Wag 1) = p(2n pe Oo? Woe ie ° ' (7) 

By putting p=3 in equation (7) we see that the solution 
of (6) is 

pr : il 


o=— FS 


% - Qi” 6(n-+3) Gos PrP uc (8) 


This gives 


yee a 
= ditt. eatin 
and two similar expressions for v and w. 
Also 
A=y"¢ 


Kpr 
=— Ss <0 Ae 
zi 2 Ye, Qn41- (10) 


Forces in opposing Distortion of an Elastic Sphere. 441 
If there are no surface tractions one of the boundary con- 
ditions is 


AwA + po Fire ar > aa UE ae (11) 


where 
C=autyvtcw 


Kpr? n+A4. 
meee 2 6(n +3)° € Qnit- 


The terms contributed to the left-hand side of (11) by the 
lil integrals we have just found are 


n+4 9 
#8 oF Send Mr Qu tH GG 43) Syl ants) 


n+2 
Bee 2 3Q, 41) f 


Ik 
Tn Roe Xen { rive Qua +3 er 2 (7°Qus1) | 


ame K 50 n 
"ears { +n) rQuirt gurree oan 


—— ber. Qn aes eS ‘(Be 
XV+2u ten OFM) 3 (23° —/ Ga) 


we 1 
+3 ur 


Now we want to make every term in the preceding ex- 
pression into a spherical solid harmonic, and since it is the 
surface value of the expression that occurs in the boundary 
condition we may put a (the value of r at the surface) for 
wherever we choose. 

In solid harmonics the pene ch expression becomes 


_. Kpa oe {(Rt4 + 2 Dart Ate ponts C) ae 
2 2n+3 3 )a In+3 ok je) : 


In the case of tidal forces or “ centrifugal force” the only 
significant part of K is — . With this value of K the above 


expression becomes 


10 n /.9n+5 n 
Se, 4 a Os Q eee +6, | aes 2 (Ss) 1, 


4492 Lifect of Radial Forces in opposing Distortion. 
where 
op | ON Ae ye 


a oe 
Se NE Oe. Qa 
2) a pager 2 
On Snake 


The above two terms should replace two similar terms given 
in the boundary condition in Love (Art. 177). The values 
of the coefficients differ, however, from those in Love's 
work. 

In order to express the functions Q,+; in terms of the 
potentials of the given forces it is necessary to equate the 
radial displacements found in the problem, measured from 
the mean sphere, to the assumed displacements, viz. 2e,Qn41. 


; be tl 
Now the radial displacement 1s a fau+yv+zw\, and the 


part contributed to this by the differential terms of the radial 
forces 1s 
» AOR n+4 Mae 
Noh) EOS) 


The value of this at the surface, where r=a, 1s 


pty nt 
N+2n 6(n+3) 


Eni. 


The corresponding terms in Love (Art. 177) are 


(A+ 3Ha?)de,Q,41- 
where 


gp 


je fpae, jill) 
10 a(A + 2p) 


ewe ON+ bp Ha? 
3N+ 2u 


Substituting these values for A and H the above terms 
become 
FeUe) Ha Ae 
NE SONI FenQnty 
which again differs from the expression I have found above. 
The error here arises from the same misconception concerning 


Note on the preceding Paper. 443 


the part played by radial forces as that from which the error 
in the boundary conditions arises. 

The rest of the work is the same as in Love’s book, and thus 
it is unnecessary to give it here. 

The erroneous theory and my theory give exactly the same 
results for an incompressible sphere, but there is a difference 
in the results for a compressible solid. I will give here the 
correct result when a disturbing force, which has a potential 


W,, acts on a sphere and the value of the ratio i is unity. 
The radial displacement is 


= 2258 W, 
= e275 + (992)9 “Gg 


where 
Sele, 
fog 
The result given by the incorrect theory (Love’s ‘ Elasticity,’ 
Art. 183) is 
2253 We 
275 +933 9 © 


6Q.= 


a 


XL Wore on the preceding Paper. By A. HE. H. Love. 
Rk. PRESCOTT’S criticism of my solution is to the 


effect that I have not used correct expressions for 
the body forces. The right way to obtain expressions for the 
components of the body force at a point, say P, is to express 
the potential at Pin terms of the coordinates of P and differ- 
entiate the expression so obtained with respect to the co- 
ordinates of P. What Mr. Prescott does is to differentiate 
(with respect to the coordinates of P) the potential at that 
point Q to which the particle that was initially at P is 
displaced. I do not know of any justification for this 
procedure. 

Mr. Prescott’s argument in the paragraph of his paper 
beginning “ Now there is a second meaning to 2, y, z”’ seems 
to me to be unsound. Whenever, as in this problem, it is 
necessary to distinguish the forces that act upon the body in 
the strained and unstrained states, the coordinates «, y, z that 
occur in the equations of equilibrium must be taken to be the 


* Communicated by the Author. 


444 Note on the preceding Paper. 


coordinates of a point of the body when in the state in which it, 
is held by the forces, not those of the same particle of the body 
when in the state that it would have if the forces were not 
acting. ‘This meaning is shown at once to be correct by 
examining the proof of the equations of equilibrium as given, 
for example, in the second edition of my book on ‘ Elasticity,’ 
Articles 44, 54. 

The incriminated solution was given in the first edition of 
my book and omitted from the second edition, because 
in the meantime I had found that it was unsatisfactory. 
The problem is concerned with an elastic solid body held 
in a nearly spherical shape by its own gravitation and 
by external forces.. The nearly spherical body is taken to 
represent the Earth. The type of the external forces 
is such as to include tide-generating force as a particular 
example. In the solution in question it is assumed that the 
stress by which the self-gravitation of the body, supposed 
truly spherical, is balanced throughout the body, is cor- 
related, according to Hooke’s Law, witha state of strain, and 
that this strain can be expressed by means of a displacement 
according to the ordinary method of the theory of Hlasticity. 
According to this method the body is regarded as capable of 
existing in two states : the first, asphere tree from gravitation, 
and therefore also free from stress ; the second, a gravitating 
sphere. The calculated displacement is that by which the 
particles would pass from their positions in the first state 
to their positions in the second state. It is essential to 
the success of the method that the strain and and dis- 
placement so calculated should be small quantities. When 
the calculation is effected it is found that, unless the 
material can be treated as incompressible, this condition is 
not satisfied. In Mr. Prescott’s notation and words, the dis- 
placement required to change 7y into 7, does alter the density 
appreciably and affects the elastic properties of the sphere. 
The assumption in regard to the nature of the stress, by 
which the self-gravitation of the sphere is balanced, is there- 
fore in general untenable, and the solution fails. Mr. Prescott’s 
would fail for the same reason even if his argument which 
is criticized above were correct. As I have pointed out in 
the second edition of my book and elsewhere, the Earth must 
be regarded asa body in a state of “initial stress.” This 
view has been advanced also by Lord Rayleigh (Proc. Roy. 
Soc. vol. Ixxvii. p. 486, 1906). The solution given in the 
first edition of my book needs correction, but not in the sense 
indicated by Mr. Prescott. 


[ 445 J 


XLIII. On the Shape of the Molecule. By R. D. KLEEMAN, 
D.Sc., B.A., Mackinnon Student of the Royal Society *. 


NHE shape of a molecule is usually assumed to be spherical 
and its diameter calculated on this supposition. One 
of the formule used is L=2No,’, where L denotes the 
mean free path of the molecule, o, the radius of its sphere of 
action, and N the number of molecules per c.c. at standard 
temperature and pressure. The value for ois usually taken 
as the diameter of the molecule. IL is usually obtained from 
experiments on diffusion or viscosity, and N froma knowledge 
of e, the electric charge on an ion. 

But the assumption that the molecule is spherical in shape 
is not admissible. Thus Meyer in his ‘ Kinetic Theory of 
Gases’ shows that the cross-section of an atom, which is 
proportional to o,?.is an additive quantity relating to the 
atoms composing the molecule. Meyer shows that this is 
only possible if the atoms of the molecale lie approximately 
ona plane. It is difficult to see how under the circumstances 
the molecule can be spherical in shape. Hspecially since 
using the values of the cross-section of molecules given by 
o,”, and the fact that the volume of an atom according to 
Traube f is proportional to the square root of the atomic 
weight, it has been shown by the writer { that the atom 
must be approximately spherical in shape. 

The object of this paper is to give a method by means of 
which direct information as to the shape of the molecule 
may be obtained. 

At the absolute zero of temperature the molecules have no 
motion of translation, and the apparent space occupied by a 
molecule or atom is its true volume. Now the densities of 
liquids at corresponding states are the same fraction of their 
density at the critical state to within a few percent. The 
densities may therefore also be said to be the same multiple 
of their densities at the absolute zero. The relative molecular 
volumes of the molecules of liquids at corresponding states 
are therefore the same as the relative volumes at the absolute 
Zero. 

Further it is probable that the relative values of o,? or oc, 
given by the above equations can only be legitimately com- 
pared at corresponding temperatures. The value of o, does 
not represent merely the real cross-section of the molecule, 
but the cross-section modified (usually increased) by the 
field of force surrounding the molecule. | 

* Communicated by the Author. 


+ Phys. Zeit. p. 667, Oct. 1909. 
t “On the Shape of the Atom,” Phil. Mag. July 1910, p. 229. 


{| 
| 


446 Dr. R. D. Kleeman on 


Now the writer has shown * that the radius of the sphere 
of action of the force of attraction of different molecules is 
the same fraction of their radius at the absolute zero. He 
had also previously shown f that the attraction between two 


ihe A ; kot et Os (ll zZ 
molecules is given by the expression (% Vm)? = o( : 8), 
~~ t Cc 


where } Ym, denotes the sum of the square roots of the 
atomic weights of the atoms composing a molecule, < is 
their distance of separation, “, 1s their distance of separation 


al 


at the critical state, and 8 =;, where T denotes temperature. 


abe Fe 
From this it follows that at corresponding states, o(-. 8) 


will be the same for all liquids, and the radii of the sphere 
of action of different molecules at corresponding temperatures 
therefore the same fraction of their radii at the absolute zero. 
We shall therefore be on much safer ground if we compare 
values of o, or o,? at corresponding states. 

Let the relative values of o,? be determined for a number 
of different molecules and also their molecular volumes in 
the liquid state, both at corresponding temperatures. Then 


if V denotes the molecular volume, the ratio ne will be the 
i 


same for all liquids if the molecules are spherical in shape, 
for V is then proportional to a4’. But if it is not spherical 
in shape the ratios are not necessarily the same, and we shall 
be nearer the truth in supposing the molecule an oblate 
spheroid of which me: is proportional to an axis of the 
1 

generating ellipse. If we denote a by o2, then Nisa fi 

i 1° 
A set of calculations of this nature has been carried out, and 
the results are contained in Table I. They correspond to 
349°7 
dd6'1 
means of the equation o,= 


T. The relative values of o, were calculated by 
mo \V? 
darn 

molecular weight, v the velocity of translation of a molecule, 
and 7 the coefficient of viscosity. In this equation, v was 


, where m denotes the 


We , J 
put proportional to i) , and then o; is proportional to 


5+ The coefficient of viscosity corresponding to the 


* Phil. Mag. p. 480, June 1910, 
f{ Phil. Mag. pp. 788-792, May 1910. 


the Shape of the Molecule. 447 
49°7 
ao0°b 
in Landolt and Bérnstein’s Tables, 5th edition. The mole- 


temperature T, was calculated from the formule given 


_ ; m i 
cular volume V is proportional to (= Yolumn 8 contains 
3 


; o o abe 
the relative values of = or ss the ratio of the axis of 


1 : 
the circular section of the spheroid to the axis at right angles 
to the section. : 


TABLE I. 
(mT)'* | Relative|Pelative| _  15,3/2 
hee 7? 107 values of ip BERGE aa sts = 
Mameof Substance. |Temp.| p. |yXx107| —. | or relative} ” |. ore or | 2m 
TT, p values of | po,” 4 GH oe 
Gis Oy. ie o) 
7 
Carbon ea 349-7 [15080 1950 |102:4| -3449 | 683-9 | 5-044 2-92 
zB 
Bihyl oxide... C,H,,0| 293-9| -7123| 7452/1039 | -4448 | 5249 | 8474 | 361 
ae C,H, | 3531] 81451057 | 95°76 -3962 | 6101 | 6-494 3-28 
Methy! Seco. | 306-21 -9858/ 997-9| 62:82 -3686~ | 4623 | 7-975 376 
aaa 
Ethyl ec) \ 343 | 830811074 |1227) -4173 | 7052 | 5-917 3:86 
5 10-2 
Chloroform ... CHCl, | 335:2|1:3935 1905 | 85-781 -3242 | 8162 | 3-972 2:66 


It will be seen that the values of this ratio are by no 
means constant, showing that the shape of the molecule is 
far from being spherical. es 

The absolute value of the ratio = for a molecule cannot 


1 

be calculated, because we do not know the absolute molecular 
volume of a molecule. But this ratio can be approximately 
obtained as follows. The writer has shown * that the diameter 
of an atom is proportional to m/*, and its cross-section there- 
fore proportional to m4, The cross-section of the molecule 
coinciding with the plane in which the atoms lie will there- 
fore be m1, and its radius is therefore proportional to 
/ 2m, Now the volume of the molecule according to 
Traube is proportional to }m?, and the length of the axis at 


mi? 
The 


(Smt) Sn 


The values of this ratio have been calculated and are given 
in the 9th column of Table I. It will be seen that they bear 


* Loe. cit. 


— 
7 
lal 
3 


AAS Dr. R. D. Kleeman on | 
approximately the same relation to one another as the 
. O-~ e e | 
relative values of — given in the 8th column. It is also 

oO 


seen that the axis at right angles to the plane in which the 
atoms lie is, as we should expect, much smaller than an axis 
in that plane. Thus it appears that the atoms of a molecule 
lie approximately in one plane. 

We should expect, however, from this that the ratio for 
ethyl propionate given in the Sth column should be greater 
than that for any “of the other substances given in the table. 
But this is not the case. Thus the atoms of an ethyl- 
propionate molecule do not lie altogether in a plane. If the 
atoms are in rotation round the centre of the molecule, we 
should expect that greater stability would be secured if, in a 
molecule containing many atoms, the latter did not lie 
exactly in a plane. 

We may therefore conclude that instead of supposing the 
molecule a sphere, we shall be nearer the truth in supposing 
it an oblate spheroid of which the ratio of the axis of the 
circular section to that at right angles to this section is 

(m3): 3/2 

ae 

The values of this ratio have been calculated tor a number 
of compounds, and are contained in Table Il. It is very 
probable that it may be possible to connect them with other 
properties of the substances such as crystalline form, and 
therefore a table of their values seems useful. 

It is of interest to inquire whether the shape of the molecule 
mv 


given by 


changes with temperature. From the equation 7= 8 

1 
we see that if the radius of the sphere of action, oj, is inde- 
pendent of the temperature, 7 is proportional to v, and 
therefore to T, since v is proportional to T’?. Experiment, 
however, shows that 7 is approximately proportional to the 
first power of the temperature, so that o, must be a function 
of the temperature. If there is no field of force surrounding 
the molecule this would indicate that the volume of the 
molecule changes with the temperature. If, however, we 
regard the molecules as centres of force, their apparent 
cross-section will change on account of the increase in the 
velocity of the colliding molecules with temperature. Max- 
well on this supposition proved that the force of attraction 
varied inversely as the fifth power of the distance from the 
centre of the molecule. We thus see that the variations of 


* Loc. cit. 


the Shape of the Molecule. 449 


n with temperature do not furnish any information as to the 
change of shape of the molecule with temperature, because 
we do not know to what extent o, depends on the field of 
force round the molecule. The same also applies to the 
variation of the coefficient of diffusion with temperature, 
because this also depends on o;, the radius of the sphere of 


action. 
TABLE I]. 
3/2 || ja, 3/2 3/2 || ja, 0/2 
(ani)? | (sm¥8)" || (antl)? | (zn8)" 
| smi? *| | Sn? || | Sm? "| | Sm? 
H. | 1415 | HOI 1276 || OO, 1712 || NH, 1-865 
oe 6f4i4 || «CCl, 1416 || N,O P7290 We Hy 2°301 
meee 1-412 || HO 1-601 1 SO, 1-726 || CH,Cl 2-023 
me, 61415 «||| HS BP On SVS eer GN 1410 
| | | C.N, | 2004 | C,H,Cl|; 3-874 
eae ee 5 
epee ene 22-1) |.) 2282 Ih Wormiie acid :24.024:. CH,O 2-116 
Ethyl alcohol......... CHO | 2802 | See 22 ee 
Pp 2 9) || Acetic acid. ......... C,H,0 2°656 
ropyl alcohol ...... €{H,0°°) “s240° *| ob an ee 2 2 
i | Propionic acid...... CLH.O 3109 
Butyl alcohol eset eeee C,H, 0 i 3 500 }] But ric acid CHO. | 3°50) 
Eeteylaleohol -.. 5H,,0 | 3975 ||, 1) lerinnicaedd CHG. | 3349 
Amy] alcohol......... O,H,,0 | 4294 | ee ee ee 
Methyl formate...... C,H,O, 2656 Todo benzene ...... C,H,1 | 3134 | 
Methyl acetate ...... C,H,O, 3109 | Bromo benzene ... C,H,Br , 3-230 
Ethyl acetate ......... C,H,O, 3505 || Chloro benzene ... O,H;Cl | 3:259 
Ethyl propionate ... C;H,,O,; 3862 | Fluor benzene ...... C,H;Fl 3°289 
Ethyl butyrate ...... OA1-O; 2S Oe ip Pentane 2-200050.:. CAL 2 oipeeeasia 
Ethyl valerianate ... C,H, ,O, SA, Octane) 22.22) 2.3.5.32. C,H. 3821 
Isobutyl butyrate... C,H,,O,| 4778 WYErOWEy oo cece. Hg ; 1-000 
Isobutyl valerianate C,H,,O, 5046 Hodme’ 222th ee is | 1416 
| | 


One point calls for remark at this place. The law of force 
surrounding a molecule is usually determined from the 
variations of the coefficient of viscosity or diffusion with 
temperature, on the assumption that the molecule behaves as 
a centre of force. But this assumption is objectionable 
since a molecule or an atom must possess an actual volume. 
The writer * has shown that if the atom is spherical in 


shape the cross-section is proportional to m™%,7.¢., if of 
2 


. . = (ox . 
represents the true cross-section of the molecule, —j, 1s 
m 


constant. This is approximately realized. On the other 
hand the chemical attraction of one atom on another is 
considerable, and must influence the apparent radius oc, of a 
molecule. Thus the writer has shown that this attraction is 


* Loe. cit. 
Phil. Mag. 8. 6. Vol. 20. No. 117. Sept. 1910. 2H 


450 Lord Rayleigh on the Finite Vibrations of a 


such that two atoms on collision may rotate round their 
centre of gravity with their previous velocity of translation, 
separated by a distance equal to their diameter. It appears, 
therefore, that the value of o, is really a complex quantity, 
and its apparent variations cannot be ascribed to the molecule 
behaving simply as a centre of force. This method of deter- 
mining the law of force surrounding a molecule is therefore 
scarcely admissible. 

Some information whether the shape of a molecule changes 
with temperature can be obtained from the following con- 
siderations. The writer has shown from the phenomena of 
surface-tension that the attraction between two molecules in 


BE asic = pe h pps 
a liquid is given by the expression 3(2m1?).K, where K 


is a constant which has the same value for all liquids at 
corresponding states and z is the distance of separation of 
the molecules. Now since corresponding temperatures are 
not equal to one another, this suggests that the dependence 
of K on temperature is not direct but indirect ;- that is, it is 
not due to a decrease of the attractive power of each atom, 
but to a change in their configuration. A change in con- 
figuration produces a change in the law of force. If the 
energy of rotation of a molecule increases with the energy 
of translation, as is usually supposed, the molecule would 
contract with rise of temperature if there is equilibrium 
between the forces of attraction and the centrifugal forces. 
It seems probable that this is what happens. 


Cambridge, May 16, 1910. 


XLIV. Note on the Finite Vibrations of a System about a 
Configuration of Equilibrium. By Lord Rayuricu, O.MZ., 
Se 


same theory of the infinitesimal free vibrations of a system, 
depending on any number of independent coordinates, 
about a position of stable equilibrium has long been familiar. 
In my book on the ‘ Theory of Sound’ (2nd ed. vol. ii. p. 480) 
I have shown how to continue the approximation when the 
motion can no longer be regarded as extremely small, and 
the following conclusions were arrived at :— 
(a) The solution obtained by this process is periodic, and 
the frequency is an even function of the amplitude (H,) of 
the principal term (H, cos zt). 


* Communicated by the Author. 


System about a Conjiguration of Equilibrium. 451 


(b) The Fourier series expressive of each coordinate con- 
tains cosines only, without sines, of the multiples of né. 
Thus the whole system comes to rest at the same moment of 
time, e.g. t=0, and then retraces its course. 

(c) The coefficient of cos rnt in the series for any co- 
ordinate is of the rth order (at least) in the amplitude (H,) 
of the principal term. For example, the series of the third 
approximation, in which higher powers of H, than H,? are 
neglected, stop at cos dnt. 

(d) There are as many types of solution as degrees of 
freedom ; but, it need hardly be said, the various solutions 
are not superposable. 

One important reservation (it was added) has yet to be 
made. It has been assumed that all the factors, such as 
(co —4n?a,)*, are finite, that is, that no coincidence occurs 
between an harmonic of the actual frequency and the natural 
frequency of some other mode of infinitesimal vibration. 
Otherwise, some of the coefficients, originally assumed to be 
subordinate, become infinite, and the approximation breaks 
down. 

I have lately had occasion to consider more closely what 
happens in these exceptional cases ; and I propose to take as 
an example a system with two degrees of freedom, so con- 
stituted that the frequencies of infinitesimal vibration are 
exactly as 2:1. In the absence of dissipative and of im- 
pressed forces, everything may be expressed by means of the 
functions 'l' and V, representing the kinetic and potential 
energies. In the case of infinitely small motion in the 
neighbourhood of the configuration of equilibrium, T and V 
reduce themselves to quadratic functions of the velocities 
and displacements with constant coefficients, and by a suit- 
able choice of coordinates the terms involving products of the 
several coordinates may be made to disappear. Even though 
we intend to include terms of higher order, we may still 
avail ourselves of this simplification, choosing as coordinates 
those which have the property of reducing the terms of the 
second order to sums ef squares. We will further suppose 
that T is completely expressed as a sum of squares of the 
velocities with constant coefficients, a case which will include 
the vibrations of a particle moving in two dimensions about 
a place of equilibrium. We may then write 


T= 4a; by? + dap $,”, aa, a aiotayt el) 
V=4aO7+4e 62+ V3+..-- - + (2) 


* See below. 


2H 2 


A452 Lord Rayleigh on the Finite Vibrations of a 


where 


Vs=m Gite br bots bi bo +Yab2 » + (8) 
giving as Lagrange’s equations 
ay, dg, /dt? + ¢, 6, + 871 62 + 2y2 b1 bo +93 GX =0, . (A) 
Ay A? ho/dt? + ey hy + 84 hy” + 273 b1 bo +72 GP =0. . (5) 
To satisfy these equations we assume 
$= Hy) +H, cos nt + H, cos 2nt + Hg cos 3nt+... . (6) 
y= Ky + K, cos nt + Ky cos 2nt + Kz cos 3nt+... . (7) 
In general we may take as one approximate solution 


ob, =H, cos nt, ee «aOR ee (8) 
with | 


n = C/ay 5 . : ° ° . ° (9) 


and in proceeding to a second approximation we may regard 
all the other coefficients as small relatively to Hy. On this 
supposition the 4th and 5th terms in (4) may be omitted, 
so that $, is separated from ¢2. Substituting from (6) and 
equating the terms containing the various multiples of nt, 
we get 

¢ Hot+3y H’=0, 


(ce —n’a,)H, = 0, 
(¢: —4n?a,)H,+ 39, Hy’=0 ; 


so that 
See by, H,’ 9 
od,=— Be, + H, cos nt— SET cos 2nt, . ine 
with 


C= WG, 
as in the first approximation. In like manner 
co Ky +4y. H,’?=0, 
(¢,— naz) K,=0, 
(cg —4n?az) Ky +42 Hy’?=0. 
Thus, if c, differs both from n?a, and from 4n?a,, we have 


yo Hi? yo Hi" 9 
© — en oS oe =i Ca Cos Pi be ° e sl 
P2 2G, 2(¢g— 4n?aq) i CT) 


System about a Configuration of Equilibrium. 453 
But if 


Co— n'y = 0, 
the inference that K,=0 does not follow ; and if 
co—An?a, = 0, 


the terms in cos2né in (10), (11) assume infinite values. 
Accordingly these two cases demand further consideration. 
We will commence with that where 


Co— nd, = 0, 
that is, where both modes of infinitesimal vibration have 
the same frequency. | 
We must now discard the supposition that ¢.=0 approxi- 
mately and be prepared to allow Ky, as well as Hy, to be 
quantities of the first order of smallness. The other coeffi- 
cients in (6), (7) are stili of the second order at least. Sub- 
stituting in (4), (5) and retaining only terms not above the 
second order, we get 
— ¢ Hy + (4 —n?a,)H, cos nt + (c: —4n7a,)H, cos 2nt+... 
+ 3, H,? cos?’ nt + 2. H, K, cos? nt +3 K,? cos? nt=0, 
Cg Ko + (¢g—n7az) K, cos nt + (co —4n7a,) Ky cos 2nt+... 
+ 3y, Ky? cos? nt + 2y3 Hy K, cos? nt + y2 Hy? cos’ nt =0; 
whence 
Cy Hot+3(3 H,?+ 272 13% Ki +43 K,’)=0, - (12) 


C2 Ko + 3(8y, Ky? +273 Hy Ki +9, Hy?)=0, . (13) 
(c;—4n?a,)H,+4(3y, Hy? + 2y2 Hi Ky t+y3 Ky?)=0, . (14) 
(¢g—4n?ay) Ke + 3(3ry4 Ky? + 273 Hy Ky +y2Hy?)=0. ~. (15) 

Also 
H., &c., K;, &e! = 0. 


These equations, arising from the terms independent of ¢ and 
proportional to cos 2at, cos dnt, &c., determine Hy, Ko, Ha, 
K,, &c. when H,, K,, and » are known. The term in cos nt 
gives further 


H, (cy — na) ==(), Ky (¢,.— nay) =0. 
Thus when : 


tei eeieeg neal th aes) Way oi CLE) 


K, as well as H, is an arbitrary quantity of the first order. 
And this completes the solution to the second approxi- 
‘mation. 


454 Lord Rayleigh on the Finite Vibrations of a 


When the process is pursued to the next stage, the ratio 
H,/K, may become determinate. In illustration of this let 
us suppose that V is an even function of both @¢, and dy. 
Thus V;=0, and 


Vi=6, 6+ 63 br: 6° +85ho%. . . . (17) 
Using this as before, we obtain 

fHj=0, K,=0, H,=0; K,=0) By &cl=0) ieee 

To determine H;, K; we have 
(¢;— 9n?a,)H3+6, Hy? +306; H, Ky?=0,. . (18) 
(¢,— 9n?a)K3 +6; K,?-+403 K, H?=0. . . (19) 

Also from the terms in cos nt 
H, [e,—n?a, +36, H,?+36; K,?]=0, . . (20) 
~ Ky [e.—n?a. +38; Ky? +36; H,?|=0. . . (21) 


EKquations (20), (21) can be satisfied by supposing either 
H, or K, to vanish while the other remains finite. Thus if 


H,=0, (20) is satisfied and (21) gives 
Cg—n 29+ 383 H,/’=0, . ° . . (22) 
determining x. From (19) we see that in this case K;=0, 
while H; is given by (18) with K,=0. 
There is also another solution in which both H, and K, 
are finite. Since by supposition 
C9/a. = 4/4, 
26 H,’?+6; Ke7 on ¢;— n'a, ae Cy (23) 


Os lalie + 20: Ky Cilaa Ny C9 be 


which determines K,?/H,’, and then either (20) or (21) gives 
n’, Equations (18), (19) determine He K, with two alter- 
natives accor ae to the sign of K,/H, 

In certain cases the ratio K,/H, Han remain arbitrary ; 
for example, if 


C= 6), able oye ic0 os 
making V, a complete square. 
The other class of cases demanding further examination 


arises when 
Cy/dg = Ac Jay. .). °. sn 


and it requires that K, should be treated as a quantity of the 


~ 


System about a Configuration of Equilibrium. 455 


first order as well as H,, the remaining coefficients bein 
still of the second order. The substitution of (6), (7) in (4), 
(5) then gives 


¢; Hy + (c, —n7a,) H, cos nt + (¢, —4n7a,) Hy cos 2nt +... 
+39, H2 cos? nt + 2y2 Hy Ky cos nt cos 2nt 
ays Ko cos? 2nt.= Of ys wi (23) 
cy Ko + (¢9—n7az) Ky cos nt + (c2— 47a) Ky cos 2nt+... 
+ 3y4 K,? cos? 2nt + 273 H, Ky cos nt cos 2nt 
aye beh? COS7 M6 =a Di ue /oys) asf (20) 
From the terms independent of t we get 
2c, Ho +39: Hi? +73 Ke?=0, 2c Ko +72 H+ 3y, Ko?=0; . 
from the terms in 3nt 
(c,—9n?a,)H3+y2 H, Ke =0, (co—9n?a.)K3 +72 H, Ko=0; 
from the terms in 4nt 
(¢;—16n7a,)H,+43y3 Ky?=0, (co —16n?a_) Ky=3ry, K,?. 


while coefficients with higher suffixes than 4 vanish. 
Further, from the terms in nt, 2nt | 


(c,—n7a,) Hy + Y2 H, kK = 0, (Cc. — nd) Ky “A Ve HH, KS = 0, 
(¢,—4n7a,)H, +39, Hy? =0, (¢2—4n7a2) Ko + $y. HY =0. 
These equations determine Ho, Ky, K,, H., Hs, Kz, Hy, K, 
as functions of H, and K, of the second order, when n is 
known. To find » and the ratio K,/H, we have the first 


equation of (30) and the second of (31). With regard to 
(24) these may be written 


Gia Wag yy ie ay 0c) eee ae) 
: (c1—n*a,)Ko+3y2Hy?=0; . . . (34) 
Les 


of which the first may be considered to determine n. Hlimi- 
nating (¢;—7n7a,), we get 


Rig f Ey ssh /Ceaf2ea elie ios nile (BB) 


This completes the solution to the second order of small 
quantities. 

If V,=0, the above solution reduces itself to that of the 
first approximation. In this case, especially if V is an even 


(27) 


. (28) 


(29) 


(30) 
(31) 


456 Prof. Barton and Mr. Ebblewhite on 


function of d, and gs, see (17), a solution, correct to the 
third order of small quantities, is readily developed ; it is 
hardly necessary to give the details. 

In ‘Theory of Sound’ (doc. cit.) I remarked upon the 
failure of the simple theory to deal with the apparently 
simple problem of the vibrations in one dimension of a 
column of gas, obeying Boyle’s law, and contained in a 
cylindrical tube with stopped ends. So far as I am able to 
see, the present extension does not help the matter. In this 
case there are an infinite number of coincidences between 
natural frequencies of infinitesimal vibration and harmonies 
of the fundamental vibration. From what we know of the 
behaviour of progressive waves of finite amplitude, it is 
perhaps not surprising if no solutions exist of the character 
contemplated. Probably after the lapse of a finite time 
discontinuity will ensue. 


Terling Place, Witham, Aug. 8. 


XLV. Vibration Curves of Violin Bridge and Strings. By 
Epwin H. Barton, D.Se., F.R.S.E., Professor of Experi- 
mental Physics, and Tuos. F. Esstewuite, B.Sc., Heymann 
Exhibitioner, University College, Nottingham *. 


[Plates VIL-IX.1 
OLLOWING the work of one of us and others+, the 


present paper deals with the simultaneous motions 

of a violin bridge and the strings. ‘The violin being mounted 
horizontally in the usual position of playing, the vertical 
motions of each upper corner of the bridge are recorded, 
also the motions of the same corners lengthwise of the 
strings. Hach string is dealt with separately and its 
vibration simultaneously with that of the bridge recorded 
photographically. Further, the strings were excited at 
various places by bowing, plucking, striking, &c., seventy- 
two photographic records being now presented. 

Comparing this work with that on the violin belly (August 
1909), the following points may be noted here :— 

1. The vibrations of the bridge show more variety of form 
than those of the belly. 


* Communicated by the Authors. 
+ Phil. Mag. July 1905, pp. 149-157; Dec. 1906, pp. 576-578; 
April 1907, pp. 446-452 ; Aug. 1909, pp. 233-240. 


Vibration Curves of Violin Bridge and Strings. 457 


2. Each of the strings gives considerable motion to the 
bridge in spite of the presence of the other three strings which 
are not sounding, whereas previously only the G-string 
gave appreciable motion to the part of the belly under 
examination. 

3. The distribution of the bridge’s motion seems to be 
somewhat intricate. Thus a certain corner of the bridge 
would scarcely stir in one direction under the influence 
of a given string excited in a certain manner, but went 
extremely well under the influence of another string or 
the same string excited differently. These circumstances 
may be due to the asymmetry of the stresses and structure 
of the violin. 

4, In some cases the motion of the bridge lengthwise of 
the string shows a frequency double that of the string, 
although such a phenomenon had not been detected by the 
previous work on the belly. 

This point has some interest in connexion with the work of 


Mr. J. W. Giltay and Prof. M. de Haas * of Delft. 


EXPERIMENTAL ARRANGEMENTS. 


The mounting of the violin was all through as in fig. 2 of 
Plate iv., Phil. Mag. Aug. 1909. The optical arrangements 
were also on the same general lines as before, so that the 
string’s motion is recorded in the positive print by a dark 
line on a light ground, the motion of the bridge being 
recorded below it by a white line on the dark ground. It 
should be noted here also that great steadiness of the arc 
light was obtained by the use of a Leitz Lilliput are lamp in 
which the positive carbon is horizontal. The plates used 
throughout were the Barnet Orthochromatic, Fast, Backed, 
and were developed by the Barnet formula. 

Vertical Motions of the Bridge——For the vertical motions 
of the H-string corner of the bridge the arrangements were 
almost precisely those shown in figs. 1-5, Plate iv. of the 
last paper (Phil. Mag. Aug. 1909), the only essential 
modification being that the leg of the optical lever m rested 
on the bridge instead of on the belly. 

For the vertical motions of the G-string corner the 
arrangements were of the same nature, but the other side 
of the violin was turned towards the lamp and plate. 


* “Qn the Motion of the Bridge of the Vioiin,” Konink. Akad. v. 
Wetenschappen, Amsterdam, Proc. xii. pp. 518-524, Jan. 26, 1910. 
Also Science Abstracts, No. 404, Mar. 1910. 


458 Prof. Barton and Mr. Ebblewhite on 


The respective magnifications in the two cases were as 
follows :— 


Actual Magnifications 
| on quarter-plate negative. 
Part of Bridge | Ratio of 
under | Magnifications 
Examination. String’s Bridge’s on any print. 
Motion. | Motion. 
| H-string Corner ......... | 2:2 5983 271. 
| G-string Corner ......... 21 Weedege 65 287 


Longitudinal Motion of Bridge.—For the motions of the 
H-string corner of the bridge lengthwise of the strings, 
the arrangements adopted are shown diagrammatically in 
fig. 1 (Pl. VII.) and in perspective from a photograph in 
mee (Pl. VI0.). | 

Referring to fig. 1, the are lamp and condenser are indi- 
cated by A L, and C, from which part of the beam passes 
direct to H, a screen pierced by a small hole. The light 
from H reaches a concave mirror m on the optical lever, and 
is thence converged to the point R on the sensitive plate PP, 
which is shot along the rails INA in the dark room. This 
gives on the lower part of the plate a record of the bridge’s 
motion, which in the positive prints appears as a white line 
on a dark ground. | 

Let us now follow the other part of the beam from the 
arc lamp. ‘Thisis reflected by the pair of plane mirrors M, M, 
to the vertical slit 8, from which the light falls upon the 
lens L,, which focusses a real image of the slit on the string 
in use at 12 or 13 cm. from the nut, 32 cm. being the 
iength of the vibrating portion of the strings from nut 
to bridge. The light is then reflected by the plane mirror M 
to the lens Lj, which gives on the plate at R a real image of 
the slit crossed by the shadow of the string; it accordingly 
leaves a record of the string’s motion, which in the positive 
prints is a dark line on a light ground. This image is 
or the upper part of the plate, precisely over that from the 
optical lever m. Thus a comparison of the two records 
gives information as to the relative phases of the vibrations 
of string and bridge. 

Let us now examine more closely how the bridge’s motion is 
recorded. Fig. 3 (Pl. VII.) shows ona larger scale the bridge 


of the violin with the optical lever somewhat out of place, 


Vibration Curves of Violin Bridge and Strings. 459 


for clearness sake. The clamp CC is fixed at a corner 
of the violin which is specially free from vibrations, and 
carries the adjustable bracket BBBBB, at the end of which 
is the table T. This table has the hole H and the slot 8 for 
carrying the two upper conical feet of the optical lever, thus 
forming its horizontal axis. The lower foot F of the lever, 
when held in position by an elastic band, presses on the 
plane P of the bridge, P being a piece of microscope-cover 
glass fastened on with shellac. 

Magnification of Bridge’s Longitudinal Motion—As the 
optical lever is now used in an oblique position, the magnifi- 
cation requires special examination. Referring again to 
fig. 1, we see that as the bridge moves lengthwise of the 
string the lever is made to rotate about the axis mA through 
a small angle dg say. Thus the normal mN to the mirror 
changes to the position mN' where N’ is above N and not 
shown in the diagram. But since the incident beam [Hm is 
at rest when N rises to N’, the reflected image R will rise 
to R’ say where IN’R’ is a straight line. Now it was 


previously shown (Phil. Mag. April 1907, p. 448) that 


RRA iby e2dan 2 1 
NN’/AN ~ tana—tan @” sag ate Mh (0. 


where a is the angle AmN and @ is the angle NmR. 
But the angle 


BEN gee eT Se AE TM ha 


where h is the displacement of the foot F of the optical lever 
and / is its distance from a line through the two upper feet 
which fit in the hole and slot and so form its axis. 

For the case shown where the E-string corner of the 
bridge is under test, the dimensions were as follows :— 


Paka by ome..., a= L672) 0 == 2° 33"... l= O68 em. 


Thus the linear magnification, on the nevative, of the 
bridge’s motion is given by 
EE ay 2 tan a 14 
eI aa aa 9 RN Pe SR J ae . — 182°]. 
h 1 *tana—tan@~ 0-68” sa etaarcabe 
For the same arrangement the string’s motion was mag- 
nified 2°1 times. Hence the relative magnification which is 
the same in any reproduction is 86°7. 
For the G-string corner of the bridge, all the arrangements 
were similar but reversed laterally and a new optical lever 


ee eee eee ae a Ae ae 


460 Prof. Barton and Mr. Ebblewhite on 


used. The dimensions and inagnifications were then as 
follows :— 


P= 13 ems., «= 11° 27’, OC = (° ogy b= Oem 
BU MR 2 tan a 13 


Minho — 99 — mike 
h { * tana—tan@ 0-68 %° 9 = Oda 


The string’s motion was magnified twice ; hence the ratio 
of the two or the relative magnification, which is undisturbed 
by reproduction, is 50°5. 

The value 101-1 of the magnification of the bridge’s motion 
was substantially confirmed by introduction of a fragment of 
blown glass under the foot of the optical lever, the consequent 
displacement of the spot being measured and the thickness of 
the glass determined by a spherometer. 

The foot of the optical lever which touched the plane on 
the bridge was made hemispherical on an oilstone, the work 
being tested bya microscope. In like manner the other two 
feet, which rested in the hole and slot, were made truly 
conical. 

As to the interpretation of the abscissee and ordinates of 
the curves on the prints, 


(1) the beginning of the time is in all cases at the left 


side ; 

(2) the ordinates for the string’s motion are always in- 
verted, 2.e. an upward motion of the shadow of the 
string means a motion of the string towards the belly 
of the violin ; 

(3) the ordinates for the vertical motion of the bridge are 
also inverted ; and 

(4) for the longitudinal motions of the bridge an upward 
ordinate denotes a displacement of the bridge towards 
the “ nut,” 2. e. towards the peg-box of the violin. 


Asymmetry of the Violin.—The necessity for dealing with 
both the upper corners of the violin bridge suggested itself 
to us on account of the well-known asymmetry of such 
instruments ; for the strings are lower in pitch, and therefore 
slacker, on one side and tighter on the other, the pitches 
being respectively g, d', a’, e'’ on Helmholtz’s notation. Thus 
the G-string side of the bridge has less pressure from the 
string on its upper edge and less where its foot touches 
the belly than is the case for the H-string side. Further, 
because of this asymmetry of the tensions and pressures 
of the strings, the body of the violin is asymmetrical also. 

Fig. 4, reproduced from a photograph of the Old English 


Vibration Curves of Violin Bridge and Strings. 461 


violin in use, shows by chalk-marks on the belly the positions 
of (1) the bass bar and (2) the sound-post. The former is a 
little bar of wood glued inside the belly and lying under the 
space between the G and D strings and extending almost 
the whole length of the belly. The position of the sound- 
post is shown by the circle under the E string and just below 
the corresponding foot of the bridge. The sound-post is a 
little straight piece of wood extending from the belly to the 
back of the violin, and held in place simply by the pressure and 
consequent friction. In this view the optical lever is again 
shown slightly displaced and tied to the strings, the adjustable 
bracket being still in position. 


RESULTS. 


The results obtained are exhibited in Pls. VIII. & IX. 
by the 72 photographs, which naturally fall into four groups 
or series according to the motion of. the bridge concerned 
and the corner of it under examination. ‘The violin was the 
one used in the former paper, and the strings were in this 
work used ‘“‘ open” and always at concert pitch. The cir- 
cumstances under which each photographic record was 
obtained are briefly indicated at the margin against each 
print. The place of excitation of the string is indicated by 
the fraction of its length from the bridge at which it was 
bowed, plucked, or struck. ‘‘ Plucked” indicates plucked 
with the finger-tip ; ‘struck”’ indicates the use of a pencil 
with several thicknesses of washleather, to form a soft pad ; 
the plectrum referred to as sometimes used in plucking was 
simply a pencil-point. The “ wooden hammer ” referred to 
is simply the pencil without the pad. ‘The strings not in use 
were still in position and at concert pitch. ‘lhey were in 
their equilibrium position if behind that in use, but if in 
front they were tied down to the neck as though being 
stopped by the fingers as in playing. We may now note 
various special points in connexion with some of the indi- 
vidual prints. 

Vertical Motion of E-string corner of Bridge. (Upper part 
of Pl. VIII.)—Figs. 1 and 2 show the G-string’s motion to be 
of small vertical amplitude but with Fourier’s series to 
infinity, the bridge’s motion being of very large amplitude 
and of quite different character. The two show by their 
general similarity the satisfactory working of the experi- 
mental arrangements, which remarks apply likewise to the 
next three figures (3-5) ; here, however, the curves for the 
string are rounded, as the forcing of the bow is absent, 


| 


462 Prof. Barton and Mr. Ebbiewhite on 


Figs. 6 and 7, on the other hand, where the harsher treat- 
ment is used, show a marked difference in behaviour of both 
string and bridge. It is noticeable that figs. 1-11 all deal 
with the motions of a corner of the bridge due to the strings 
most remote from it. The other strings when bowed or 
plucked scarcely move this corner vertically, but did so to a 
slight extent if struck. These results are given in figs. 12-14. 

Vertical Motion of G-string corner of Bridge. (Lower part 
of Pl. VIII.) —Figs. 15-17, for the G-string bowed, by their 
close similarity, give another proof of the satisfactory working 
of the apparatus. The string has again only a moderate 
vertical amplitude and shows the two-step zigzag, while 
the bridge’s motion is large and of striking character. 
Figs. 18-23, in which the G-string is struck or plucked, 
again show the rounded curves instead of the sharp peaks 
always present when the string is well bowed. 

With this corner of the bridge it was again found that no 
considerable vertical motion could be produced by any but 
the G-string, the E string giving nothing appreciable. 


The D and A strings were also ineffeetive when plucked. | 


Figs. 24-26 give the best results obtained for the D-string, 
and figs. 27 and 28 for the A-string which was only effective 
when bowed. 

Longitudinal Motion of E-string corner of Bridge. (Upper 
part of Pl. [X.)—Figs. 29 and 30 are not, like most of the 
others, true displacement-time curves, because we here failed 
to secure a simple vertical motion of the spot of light on the 
plate. Hence the abscissee which should denote time simply 
are here complicated by the horizontal component of the spot’s 
motion. In other words, the axes are now oblique instead of, 
as usual, rectangular. Figs. 32 and 33, showing the effects 
of bowing the A-string, are closely analogous to figs, 27 and 28 
produced by the same excitation. Fig. 37, in which the 
A-string is plucked, shows a striking vibrational curve much 
slower than that ot the string and apparently due to some 
vibration of the instrument us a whole. This is interesting 
in the light of a similar phenomenon noticed recently by 
Mr. G. H. Berry in work on the sound-board of the piano- 
forte (Phil. Mag. April 1910). Other examples of this slow 
vibration in the present work occur in figs. 31, 50, 60, 66, 
and 71. 

Figs. 38 and 39, in which the D-string was bowed, show a 
moderate motion of the bridge though the string is scarcely 
moving vertically. It may, however, have had a fair hori- 
zontal motion, which is of course unrecorded by the photo- 
graph ; indeed it is obviously difficult to produce by bowing 


Vibration Curves of Violin Bridge and Strings. 468 


much vertical motion of the middle strings (D and A). The 
motions of the bridge in these prints are seen to be highly 
complicated ; the traces are, however, true displacement-time 
curves. 

Figs. 40 and 42 show similar complicated motions of the 
D-string when plucked at a seventh; figs. 41 and 43 also 
forming a second but different pair for plucking at one- 
fourth. It may be noted that the D is the thickest of the 
four strings. 

In figs. 44 and 45 we have the G-string bowed, and 
so obtain again the characteristic two-step zigzag. The 
bridge’s motions are here represented by traces which, though 
so strange, are true displacement-time curves and show a 
resemblance to the other bowed sets of this series, figs. 29 
and 30, 32 and 33, 38 and 39. 

The remaining figures of this series deal with the G-string 
plucked or struck. Of these, fig. 47 shows the bridge’s 
motion to have the pitch of the string with very little appear- 
ance of the presence of the octave. In figs 46 and 50 the 
octave is quite distinct in the bridge’s motion; while in 
figs. 48 and 49 the octave is paramount, the fundamental not 
being evident at all. In this last respect these two are unique 
among all the prints hitherto obtained. An approach to this 
state of things is however seen in various other cases, as, for 
example, in figs. 44 and 45. 

Longitudinal Motion of G-string corner of Bridge. (Lower 
part of Pl. [X.)—Figs. 51 and 52, for the G-string bowed, 
do not give pure displacement-time curves for the bridge’s 
motion, as here again we did not succeed in securing a purely 
vertical motion of the spot of light on the plate. Figs. 53-57 
deal with the G-string struck or plucked, and call for no 
special remark. 

Figs. 58 and 59,for the D-string bowed, are again affected 
by an oblique motion of the spot of light on the plate. The 
next figure, showing a superposed slow motion, has already 
been referred to. Tig. 62 shows, by accident, the very 
beginnings of the vibrations of the D-string plucked and 
the consequent motion of the bridge. The plate was 
unintentionally shot rather tov soon; hence the greater 
part of it shows only the string and bridge at rest in the 
displaced or drawn-aside positions due to plucking, the 
initial motions of each on letting go being also seen at 
the right margin of the print. 

In figs. 64 and 65 the curves for the bridge’s motion are 
again complicated by a slight sideways motion of the spot 
of light on the plate. Figs. 66-68 show the very small 


A64 Mr. J. J. Lonsdale on the Ionization 


longitudinal motions of the bridge obtainable by striking or 
plucking the A-string. Fig. 69, with a smaller motion of 
the string excited by the plectrum, gives a larger motion 
of the bridge. The last three figures (70-72) show the only 
appreciable motions obtainable for this corner of the bridge 
by exciting the H-string. Fig. 72 is very remarkable in 
that the string’s motion, though entirely vertical, leaves so 
small a record, and yet the longitudinal motion of the other 
corner of the bridge is quite considerable. 


Univ. Coll., Nottingham, 
June 29th, 1910. 


XLVI. Vhe Ionization produced by the Splashing of Mercury. 
By J. J. Lonspate, M.Se.(Dunelm), B.Sc. (Lond.).* 


REVIOUS work on the ionization produced by splash- 
ing, or by bubbling gas through liquids has usually 
been carried out with liquids having considerable vapour 
pressure ¢. The results show that the ions produced generally 
move with a very small velocity, varying from a few mms. 
to ‘001 mm. per second under a potential gradient of one volt 
per cm. The question is whether these small velocities are 
due to the condensation of the vapours on the ions, or 
whether, as Bloch f suggests, the ions are of a kind altogether 
distinct from the usual Rontgen ray ions. With a view to 
gaining evidence on this point, I have investigated the 
ionization produced by splashing mercury, as there will be 
little condensation in this case. As the mechanism of the 
production of ions by the splashing process is not clear, various 
other points have received attention. 

The apparatus used is shown in fig. 1. The splash- 
chamber consisted of a wrought-iron cross-piece fitted with 
nipples, allowing the splash-plate or the upper iron tube to 
be readily removed. Mercury fell from the iron funnel A 
on to the plate B, and the ions so produced were pulled by 
an air current along two brass tubes, insulated from each 
other and carrying two insulated electrodes, C and D, placed 
axially in the tube. © and D could be connected when 
desired to a Dolezalek electrometer. The connexions were, 
of course, suitably screened by earthed conductors. With 
ebonite insulation certain irregularities were shown when 
the direction of the electric field was reversed in the space 


* Communicated by Dr. R. S. Willows. 


+ Kihler, Ann. der Physik, 1903, p. 1119. Aselmann, Ann. der 
Physik, 1906, p. 960. 
t Bloch, Compte Rendus, cxly. p. 54. 


produced by the Splashing of Mercury. 465 


surrounding O and D ; paraffin, which largely reduced these 
irregularities, was therefore used. The brass tubes were 


connected to the insulated pole of a battery, the other pole 
of which was earthed, and the ions were driven over to C 
or D, originally earthed and connected to the electrometer. 

As a preliminary investigation, it was deemed advisable 
to test if the ionization of the air was the same if the mercury 
struck the drop-plate in an electrified condition or not. If 
the dropper and the splash-plate are both earthed the mercury 
might be electrified in its passage through the air and so 
reach the plate in a charged condition. ‘To test this, the 
dropper was placed above a long narrow cylinder provided 
with a hole just large enough for the stream to go through 
without splashing. The cylinder and leads were screened by 
an earthed outer cylinder. If any charge is given to the 
mercury on passing through the air, it will be carried in to 
the cylinder, the equal and opposite charge remaining in the 
air above. The ionized air-charge and the charge on the 
plate will be of equal amounts and opposite in sign, so that 
any current given to the electrometer will be due to air 
friction. Various heights were tried but no definite evidence 
of electrification was found. 

Next the dropper was insulated on a block of wax, the 
splash-plate was earthed, and the currents were examined on 
the second electrode of the final apparatus. The current | 
was taken when the dropper was earthed. Then the dropper 
and the mercury were charged to voltages varying from 
—390 to +890. Both positive and negative currents were 
examined, but no definite variations were found in the 
ionization currents. Thus in the after-work no variation of 


Phil. Mag. 8. 6. Vol. 20. No. 117. Sept. 1910. 21 


466 Mr. J. J. Lonsdale on the Lonization 


the ionization current could be attributed to the electrification 
of the mercury. 

To ensure that no variation in the ionization was due to 
the variations in the state of the mercury, different samples 
were tried and the currents taken. If the mercury were 
purified by bubbling air through for a considerable time 
before using, the results showed no variation. 

The effect of changing the surface on which the mercury 
splashed was also investigated. Previous investigators have 
stated that the material of the plate has no effect, at least if 
the plate is wetted by the liquid. The electrometer having 
a sensitiveness of about 800 mm. scale-divisions for a volt, 
the mercury splashing on iron gave a current of 120 mm. 
in 20 seconds due to positive ions. If the splash-plate were 
re-cleaned and polished the variation was very slight. The © 
negative ionization was very much less. A voltage of about 
200 volts was put on the second electrode in both cases. 
A platinum splash-plate gave under similar conditions a 
positive current of about 30 mm. in 20 seconds. The 
negative current was scarcely measurable. When mercury 
was dropped on mercury the positive current was very 
small, often nothing; the negative current was larger. Glass 
was tried as presenting points of interest. Very variable 
currents were observed, as might have been expected, owing 
to the field produced near the source by the electrification 
of the glass. If the glass plate was used once and then 
heated to discharge it, the results were in moderate agree- 
ment and gave a positive currentof about 14 mm. in 20 seconds. 

With the apparatus used it has been shown that the 
mobility of the ions is given by * 

(b°>—a? b 
2Vt Nog. 


Uu=p 


where 0 is the radius of the outer tube; 

a is the radius of the inner electrode ; 

V is the voltage on the outer tube; 

t the time taken by the blast to pass along the length 
of the electrode ; 

p the ratio of the saturation currents on the second 
electrode. (1) when field V is on D; (2) when 
there is no potential difference between D and the 
tube. 

A first examination of the positive ions showed that the 
current gradually rose as the voltage increased and then 
remained constant between voltages of 85 and 150 volts. 


* J. J. Thomson, ‘Conduction of Electricity through Gases,’ p. 59. 


produced by the Splashing of Mercury. 467 


Taking 85 volts as saturation voltage this gives with this 
apparatus a mobility for the slowest ions of °013 cm. per 
volt-cm. for the positive ion. 

_A similar examination of the negative ionization gave the 
saturation voltage of the negative as about 300 volts, 
corresponding to a mobility of -004 cm. per volt-em. The 
fact of the negative ion having the least mobility appeared 
so remarkable that the positive current was again examined 
to see if it had really been saturated. 200 volts were placed 
on the first electrode and 200 on the second. A considerable 
current was found on the second electrode. The curve 
giving the relation between the current and the voltage was 
therefore investigated over larger ranges of voltages and had 
the appearance shown in fig. 2, curve A. A distinct flat 


part exists between voltages of about 100 to 150 volts and a 
further flat part from 300 volts onward. The upper part of 
the curve may be due to ions of small mobility or it may be 
due to neutral doublets, similar to those observed by Thomson 
in canal rays, which break up-under the action of a field or 
from other causes. If ions of small mobility were present 
we should expect the curve to rise gradually, never showing 
a flat part until all the ions are removed, and, as is seen, this 
is not the form the curve takes. It cannot be due to the 
formation of ions by collision as generally understood, as 
this would require a field of 30,000 volts per cm. ; the field 
: 212 


468 Mr. J. J. Lonsdale on the Ionization 


near the centre at the electrode was never greater than 
3900 volts per cm. 

The curve giving the relation between the current on the 
central electrode and the voltage on the outer tube when 
ions of varying mobilities are present :— 


The velocities of the ions under a gradient 
of a volt-cm. are respectively . . . . wu 


The number of ions per c.c. of each kind .  n 
Saturation voltage of each ion (ascending 


Wigs: 9 Ole ee 


aia) | 2tde a eee 


order) bs li my. asa 
The time for the air to pass the electrode . ¢ secs. 
The measure of the blast . . . . . . Bice. per. see: 


Let r be the greatest distance from which an ion can be 
driven into the electrode by a voltage V, then 


2Vtu 
5 
loge- 


2 Pia 


—gq*= 


f fe 


Then in one second the ions will be driven into the electrode 
from the volume 


B(7? —a?) 
(l?—a?) © 
If e is the charge on an ion, then the current due toa 
particular ion is 
en(7?—a?)B 
(Faa) 
The whole current due to any voltage V, when that voltage 
is less than V, the saturation voltage of the fastest ion, is 


eB 


ae ee (7? —@7)ng+ (7? —a?)n3+ ok ea (ore h, 
ea (Vuyny + Vurgng + Vugns+...... yg 
fe is 
(b?—a’) log. = 
or C = k(Vuyny + Vigne + Vusng aap 4 ¥, 


where & is constant, and as wu and are also constants this 
part of the curve is a straight line through the origin. If 
the voltage lies between V, and V, then the current 


=k(Vyuyny + Vugng+ Vugng + ...... ays 


produced by the Splashing of Mercury. 469 


This part of the curve is also a straight line. When a 
voltage above V, is reached the current will be given by 


h(Vyuyny + Vougne+ Veugngt...... ys 


where the forms Vwn are constant. This will be a straight 
line parallel to the axis of «. Thus the curve should 
gradually rise, never showing a flat part until all the ions 
are removed. The curve evidently has not this form in 
curve A in fig. 2. 

Further, if a voltage above 300 volts is put on the first 
electrode, then there should be no current on the second, 
since beyond this voltage the curve is flat. Thus a voltage 
of 342 volts should drive all the ions on to the first electrode, 
but in most cases observed (the exceptions are given later), a 
current was found on the second of such a size that if it had 
been due to free ions it should have shown on the saturation 
eurve. ‘I'he numbers below show this: they refer to 
different sets of observations with different electrometer 
sensibilities, different heights of fall, &., but the bracketed 
pairs were taken under the same conditions except that the 
field was reversed. 


V, the voltage on the first electrode ; 
V, the voltage on the second electrode ; 
e the current observed on the second electrode when the 
given voltage V was on the first ; 
C the current observed on the second electrode when the 
first was earthed. 


Wi | PANNE ¢: C | 
Fee eae | 
598 598 9 O77 | 

tai tas —25 

598 598 8 54 

Bea ned 8 ore 
| 600 | 600 7,8,7 43 | 
: — 6C0 ~ | 8,8 Small | 
: 544 ae i 5, 5 20 | 
544 — 544 6,4 0 | 


~The last resnlt is remarkable. A capacity had been 
placed in parallel with the electrometer, and thus the 


470 Mr. J. J. Lonsdale on the Ionization 


saturation current for the positive ionization was only 20 mm. 


divisions in 20 seconds. The current due to the negative 
ionization when the first electrode was earthed could not be 
observed, yet when the field was put on the first electrode 
a current of 5 divisions was found. ‘This appears to show 
that the field produces the ions from the doublets, if such are 
present. The initial leak of the electrometer could not be 
observed : it was certainly less than 1 mm. in 20 seconds. 
Examining these results, if we presume that they are due to 
slow ions, then from the dimensions of the apparatus the 
mobilities of both the positive and negative ions must be 
less than 0018 cm. per volt-em. It is worthy of note that 
no matter how largely the total current C may differ in a 
given pair of observations for the positive and negative ions, 
c is practically the same for either sign. This is what we 
should expect if the current is due to the breaking up of 
doublets. 

The saturation curve with negative ions is shown on fig. 2, 
curve B. No departure from “the usual form of saturation 
curve was found with these ions, but the smallness of the 
current may account for this. The second flat part does not 
always appear even with positive ions. Whenever this is 
missing, then all the ions can be taken out on the first 
electrode by a voltage of about 100 volts. 

The ionization currents vary with the height of the drop, 
the other conditions remaining the same. The investigation 
was made in the following way :—A definite height of drop 
being taken, the current for positive and negative ions was 
observed on the second electrode with the first earthed. 
Then a voltage of 114 volts was placed on the first electrode. 
As this voltage is on the flat part of the curve first occurring, 
the faster positives will be driven on to the first electrode. 
If the highest voltage is then placed on the second electrode, 
the ions giving rise to the upper part of the curve will be 
driven on to the electrode. The results averaged and 
reduced to a sensitiveness of the electrometer of about 
1000 mm. per volt are given below :— 


Height of dropper . . mg ot) 
Voltage on the first eeioas : 

Voltage on the second electrode . 

The total current due to negative ions . 
The total current due to positive ions 


The positive current when the voltage V, is on he 
second electrode 


1 


2 


mA << 


ss 


produced by the Splashing of Mercury. 471 


The results are plotted on fig. 3. 
Fig. 3. 


| 
ae 
| 


The numbers under p show the current on the last electrode 
with a field of about 590 volts, when the voltage V, is on — 
the first : they therefore represent the ions causing the second 
rise in the saturation curve. For convenience we will call 
these the slow positive ions. The last column shows the 
-total negative ions, and as is seen the slow positive are 
practically equal in amount to the negative ions. It, there- 
fore, the doublet theory is correct, the negative ions arise 
altogether from the breaking up of doublets. This may 
perhaps account for the current-voltage curve of the negative 
ions showing no peculiarities, as the doublets may be broken 
up in continuously increasing amount. The numbers show 
that below a height of 21:°5 cms. no slow positives are pro- 
duced, and, as we should expect from the doublet theory, the 


(9) Mr. J. J. Lonsdale on the Ionization 


negatives are also absent. In these cases the saturation 
curves show no pecaliarities. 

At the heights 14°8 and 21°5 voltages of 600 were placed 
on both electrodes, but no current could be observed on the 
last electrode. This shows that at these heights no doublets 
are formed. At a height of 6 cms. the mercury struck the 
plate without breaking into drops and no ions are produced. 
It the mercury is forced by pressure from the dropper, there 
is no ionization produced unless the mercury forms visible 
drops before striking the plate. 

Professor J. J. Thomson has asked whether ions are first 
produced by splitting doublets, or are ions first produced and 
then doublets formed by the combination of positive and 
negative ions. These results at the lower heights show that 
ions are produced by dropping mercury without accom- 
panying doublets. At greater heights both doublets and 
ions are produced. As we get the ions without doublets in 
certain cases it would seem that something more than ions is 
necessary to produce doublets. 


With a view to further information on the subject of 
doublets at atmospheric pressure, the ionization produced by 
heating aluminium phosphate and lime was examined by the 
same apparatus. The salt to be heated was placed on a thin 
piece of platinum-foil in a wide glass tube and then heated 
electrically to a bright white heat. Air filtered by cotton- 
wool was drawn over the heated salt into the apparatus. 
The curves giving the relation between the current and the 
voltage shown on fig. 2, are the currents from aluminium 
phosphate, C and D, and from lime, E and F. It will be 
seen that there is a distinct kink in each at about the same 
voltages as the mercury curve, and the saturation voltage of 
the second part is the same as the saturation voltage of the 
negatives. 


vy on If the saturation curve be examined 
314 130 in the case of aluminium phosphate, 
the current does not vary by more 
352 130 leas 
than four divisions for voltages from 
390 130 314 to 656. Yet if a voltage above 
432 130 300 is put on the first electrode, and 


ATA 128 the current examined on the second, it 
516 130 is seen to be larger than the four divi- 
558 126 sions difference observed in the current 


rs on the first, from the following numbers. 
6u0 130 The first three lines refer to lime and 
656 130 the last to aluminium phosphate. 


produced by the Splashing of Mercury. 473 


Positive saturation current on the first electrode. . P 
Negative saturation current on the first electrode . N 


Positive current obtained with a voltage of 600 on 
the first electrode, the current nis taken on the 


second electrode .. . Ne keke Na Dah. 
Negative current similarly ubeatand Bie avnea es eee Fa 
P, SOE Seal P N. n | 
/ Pee a Rete | eames 1, Ra ane Aa vy 
| 170 13, 14, 15 200 15, 15 | 
| 79 13,14 164 ai 1 B.S 
‘ | 
| 175 19, 20, 20 224 19,15, 16). | 
i H 
| 
| 114 eek. 145 | Ce pe 
| | 


The values for the positive and negative currents on the 
second electrode, with a large voltage on the first, show a 
remarkable equality despite the fact that the total positive 
and negative currents differ, although not to the same extent 
as the positive and negative currents produced by the 
splashing of mercury. Ifthe mercury were allowed to splash 
and the air drawn through a tight cotton-wool plug 2 inches 
long before reaching the testing vessel, it was found that 
ions were still present. These, again, are exceedingly slow 
ions that have passed the plug, or else they have been pro- 
duced from doublets after they have passed the plug. Similar 
results were obtained by Garret and Willows for the ions 
produced by the halogen compounds of zine (Phil. Mag. 
1904, vol. vil. p. 437). 


Summary. 


(1) Splashing mercury on an iron plate produces a large 
excess of positive ions over negative ions. 

(2) A considerable proportion of these ions have a very 
small velocity. 

(3) The amount of ionization depends on the nature of the 
surface on which the splashing is produced. 

(4) The current-voltage curves for the positive ions show 
peculiarities which may be most readily explained by 
snpposing the presence of neutral doublets, which are broken 
up by the field or other means. Below certain heights these 
doublets are not produced. 


ne 
t 
{| 
l 
i 
| 


TH 


474 Wiltshireite: a New Mineral. 


(5) Similar peculiarities in the saturation curves for the 
positive ions of aluminium phosphate and lime are also 
noticed. 

(6) The negative ions from these three sources show no 
such peculiarities. 


I have to thank Dr. R.S. Willows for the use of the 
Cass Physical Laboratory and much kindly advice and 


assistance. 


XLVII. Wiltshireite: a New Mineral. 
By Prof. W. J. Lewis, 4.A., FBS." 


URING a recent visit to Binn (Valais) I obtained, 
amongst several other interesting specimens, one which 
gives on measurement a series of angles which leave little 
doubt that it has not been hitherto described. It is associated 
with a crystal of sartorite in a cavity in the well-known 
dolomite; and from its position it is probably of more recent 
origin. Its colour is for the most part tin-white, but a few 
of the facets have a russet tarnish. The crystals are small 
but piled on one another in almost absolutely parallel 
orientation, and the same facet on separate individuals gives 
in many cases a distinct single image, although some of the 
most important cases give two images separated by some ten 
minutes. So far only a crystallographic determination can 
be made, and its chemical composition must be left until 
further specimens are found, though it is highly probable 
that it is a lead sulpharsenite. 

The faces 201, 302, 101, 001, and 101 are smooth and bright, 
and give for the most part good images; they are all very 
small end-facets. The hemi-pyramids which occur in two 
homologous zones symmetrical to a symmetry-plane are also 
smooth and bright; and single images were got from several 
distinct and separate facets on different parts of the specimen. 
The faces in the vertical zone 100, 310, 320, 120, and 010, 
are strongly striated parallel to their edges, along which the 
crystals are elongated; they give good images in zones which 
likewise include pinakoids and hemipyramids, but direct 
observations over the vertical edges give very indifferent 
measurements. 3 | 

The crystal belongs to the oblique system, and its elements 


* Communicated by the Author. 


Elster and Geitel Electrical, Dissipation Apparatus. 475 
may be given as A 
100 : 101=48° 47”3, 100 : 001=79° 16’ 
and OOP: O11 46° 25°75; - 
or by B= 79° lo and a 3b 5 ¢=17587.;1,: 1:070; 


The following are readings in important zones :— 


| Computed. Observed. 
> ae Set Tae kaa: we 
(e) 4 (o) { | 
100 : 201 33 37-7 32 36 | 
302 39 22 40 li 
101 48) ATS) 48 47 
001 79 16 79 15 
Pa 101 116 24-6 116 33 
100 : 522 ST) 33 Bein 
211 43 9 AS AU 
233 50 14 50 7 
111 coe yy 59 10 
122 FO. 1 70 O 
. 011 82 387°5 82 38 
122 96 19 9% 2 
111 108 44 108 39 
211 128 56 128 58 
522 136.11 136 2 
101 : 212 25 36 25 35 
Seca 43 466 43 45 
011: 001 46 25:75 46 24 | 
| 011: 001 46 35 | 
| 92 51:5 92 58 | 


| 011: 011 


T propose for it the name of Wiltshireite, as a token of 
respect for the late Rev. Professor Thomas Wiltshire, Hon. 
Sc. D., at one time Professor of Mineralogy in King’s College, 
London. a 1 pe 

Cambridge, 13 August, 1910. se 9 ed ] 


4, 
hi 
cy 


XLVI. Discussion of Results obtained at Kew Observatory 
with an Elster and Geitel Electrical Dissipation Apparatus 
from 1907 #0 1909. By C. Caren, Se.D., LLD., FRS* 


(From the National Physical Laboratory.) 
§1. <. ordinary Hlster and Geitel dissipation apparatus 


has been.in regular use at Kew Observatory since 
the end of 1906, and the Annual Reports for 1907, 1908, and 
1909 have contained summaries of the results derived from it. 


* Communicated by the Author. 


Pe RE 


= pashciaete -aibeestinnady aa Sa Se SSS OO SS 
Sa =o — = 6 ee 


Se SS 


_— ae RS ED 
= = 


476 Dr.C.Chree: Results obtained at Kew Observatory with 


The present paper discusses the results in a more complete 
fashion. The observations have been taken at a fixed spot 
in the Observatory garden, and the instrument has been used 
in an invariable way with the cover on, following it is believed 
the original procedure approved by Elster and Geitel. 

During the observations the instrument has stood on a 
stone pier 1:2 metres in height situated near the middle of 
the garden, at a considerable distance from any tree or 
building and freely exposed to the wind. The centre of the 
dissipation cylinder is 1:45 metres above the ground. The 
instrument has been wholly unprotected except by its own 
cover. No one is near the instrument unless when actually 
reading it, and when doing so the observer stands to leeward. 
Observations have been made only when it was dry, and when 
the wind permitted. During high winds the electroscope 
leaves are not sufficiently stationary, and after a little 
experience no observations were made under such conditions. 

All the observations utilised here were taken between 
2and 4 p.m. The regular dissipation observations were 
preceded and followed by a leakage experiment, to deter- 
mine how much of the apparent loss was due to defective 
insulation. There were usually three dissipation observations, 
each lasting 20 minutes ; the first and last being with charges 
of the same sign, the intermediate with a charge of opposite 
sign. If on one day there were two positive charge experi- 
ments, then on the next day there was only one. Thus the 
number of experiments with positive and negative charges 
was roughly equal, and the mean time of the experiments on 
the same day with the charges of opposite sign practically 
coincided. The mean monthly and annual results for the 
dissipation of positive and negative charges may thus be 
regarded as exactly corresponding to one another. 

At the time of observation, the observer noted the amount 
of cloud (scale 0 to 10) and its type or types, also whether 
the sun was shining, and if so whether brightly or otherwise. 
He further noted the state of the atmosphere, whether clear, 
hazy, misty, or foggy, and the direction of the wind. Most 
of the observations were taken by Mr. E. G. Constable, the 
senior assistant in the Meteorological department. In his 
absence they were taken by Mr. E. Boxall, who is also an 
experienced meteorological observer. For the purposes of 
the present discussion, particulars have been derived from the 
curves of the self-recording instruments as to the mean 
values of temperature, barometric pressure, relative humidity, 
and electric potential gradient during each afternoon’s 
observations. 


an Elster and Geitel Electrical Dissipation Apparatus. 477 


§ 2. The values of a, and a_, the percentage losses per 
minute of positive and negative charges, have been calculated 
from the formula 


LM RC: 1 wits 
a/100= ee ‘ +4 log Vi = SVs siya es (1) 


where V, denotes the initial, V,; the final potential in the 
dissipation experiment, V, the initial and V/’ the final potential 
in the leakage experiment, both experiments being supposed 
to last ¢ minutes, n denotes the ratio (capacity of electro- 
scope alone)/(joint capacity of dissipator and electroscope). 
Unfortunately, 7 is a quantity which it is not easy to 
determine accurately, and while high precision in the value 
of n is unimportant in the leakage term inside the long 
bracket, supposing the insulation to be good, it is of course 
important in the factor (1—n)~! outside the bracket. When 
dealing with comparative results from the same instrument, 
Elster and Geitel, in some at least of their earlier work,. 
omitted the factor 1—n, using the notation E instead of a. 
The fact that EK and a represented something more than a 
mere difference in notation was overlooked in the preparation 
of the tables of dissipation results published in the Kew 
Reports for 1907, 1908, and 1909. The values assigned there 
to a, and a_ really answer to E, and H_ in Hlster and 
Geitel’s notation, and thus presumably require multiplication 
by (1—n)~1 to be comparable with values published for 
other stations. The value obtained for n in direct experi- 
ments at Kew was 0°3, the corresponding value of (l—n)7} 
being 1/0°7 or 1°43. 

So long as the insulation is kept satisfactory the ascription 
of a wrong value to n practically alters all values of a, and 
a_in the same proportion, and so is without influence on 
any conclusions that depend only on relative values. Such 
an error is for instance without effect on values of a_/a, or 
on the annual variation observed in a_ or a4. 

§ 3. There is another question affecting the interpretation 
of the results. 

Elster and Geitel assume “ dissipation” to follow the law 


GN ide PaO 8, BOURSES OND 


This is at least approximately true in air so long as the 
potential gradient is small, a being a measure of the 
conductivity. As the gradient, however, is raised, the curve 
in which abscisse represent gradient and ordinates current 
departs markedly from a straight line, and then throughout 
a considerable gradient range remains practically parallel to 


—— a ee 


478 Dr.C. Chree: Results obtained at Kew Observatory with 


the axis of abscisse (‘‘ saturated” current condition). A 
further stage then presents itself which does not concern us. 

Mache and v. Schweidler * assert, what seems to be now 
generally allowed, that the conditions in the Hlster and Geitel 
instrument as ordinarily used are those of the “ saturated ” 
eurrent. If so, then the true formula is 


dV [di + A=) ck ese (3) 


where A is proportional to the number of ions being gener- 
ated in unit of time in unit of space surrounding the 
dissipator. This last formula leads to 


A=(V,<V) jij. oo 


where V, is the original value of the potential, and V; its 
value after time t. 

Let us compare this with the corresponding results from 
the equation assumed by Elster and Geitel, viz. 


a= (1/t) lox (Vo/V:). .. =) 
Writing this a=—(1/t) log {1+(V —Vo)/Vo}, 


we see that as a first approximation, provided (V:—V,)/V, 
be small—as is normally the case at Kew—we have 


a=(1/t) (V,—Vi)/Vo- » > ee 
The forniule (2) and (5) would apply to the ideal Hlster 


and Geitel instrument, in which the capacity of the electro- 
scope is negligible, or n=0. The result calculated from the 
right-hand side of (5) would represent a/100 in Hister and 
Geitel’s notation. We see, however, that so long as the loss 
of charge intime ¢ is small, we may to a close degree of 
approximation replace (5) by (6). If, then, the true law of 
loss of charge be not (2) but (3), the calculated value of 
a/100 is very approximately the true value of A/V,, 
and so varies approximately as A—the number of ions being 
generated in unit of time—provided V, is constant. 

In the Kew observations Vy was not really a constant, but 
it seldom departed much from its mean value, which was. 
about 180 volts for both the positive and negative charges. 

It is unquestionably desirable that no doubt should exist as 
to the exact physical significance of observational quantities. 
It is doubtful, however, whether any one of the quantities 
measured in atmospheric electricity observations is wholly 


* Die Atmospharische Elektrizitét, pp. 68, 64. 


an Elster and Geitel Electrical Dissipution Apparatus. 479 


unambiguous. And, on the other hand, there are the facts 
that Elster and Geitel instruments have had an extensive use 
on the continent, and that there is a larger mass of compara- 
tive data for a, and a_ than for any other atmospheric 
electricity element, except perhaps potential gradient. 

It has thus seemed worth while to run an Elster and Geitel 
apparatus at Kew long enough to obtain fairly representative 
data, and having done so it has appeared desirable to analyse 
the results. 

§ 4. Table I. (p. 480) gives results from the three years 
combined. N denotes the total number of days’ observations. 
a, and a_ are the percentage losses per minute as derived 
from (1) with n=0°3, t being almost invariably 20 minutes. 
a denotes the arithmetic mean (a,+a_)/2, a quantity em- 
ployed by some observers as a measure of dissipation. q repre- 
sents the arithmetic mean of individual days’ values of a_/a,, 
while g’ stands for ({a_)+(2a,), where > denotes summation 
for all days of the month. , 

Of the three seasons, Winter is composed of four months 
November to February, Summer of May to August, and 
Eguinox of the remaining four months. 

The values assigned to the year and the seasons, except for 
g', are arithmetic means from the monthly values, but in the 
case of q’ they are derived from the seasonal values of a, and 
a_ carried one significant figure further than in the table. 
Table I. gives also mean values corresponding to the times of 
the dissipation experiments for potential gradient P, screen 
temperature, barometric pressure, amount of cloud (scale 0 
to 10), and relative humidity. Meteorological conditions vary 
so much that it appeared desirable to indicate their character. 
The pier carrying the dissipation instrument and _ all 
uninsulated parts of it were at zero (the Harth’s) potential, 
and the electric field in the immediate neighbourhood was 
necessarily irregular. The values assigned to P are intended 
to refer to a spot in the open. They were derived from the 
electroyraph curves by applying factors obtained by reference 
to absolute observations with a portable electrometer. The 
mean value of the factor was 1°41. This is higher than the 
values hitherto employed, recent experiments having shown 
that previous values were too low. » 

The values of a, and a_ are considerably lower than those 
recorded at most continental stations. The largest values 
actually measured during the three years were 1°80 for a, 
and 3°41 for a_. The values obtained for g or q' are excep- 
tionally large. 


ee 


i ~4 

= 6g 6-9 980-08 0-99 91 aa 6G-T 618: 196- TL9: ee oe oe SER STONES 
= 9 $.g 916-62 1-9¢ BES 8¢-T GL-T 6F9- F6L- $0G- fet? ee ae ae xoumby 
3 GL L-9 ZSL-08 6-8 Poe IFT 29-1 Ife. $89. OGF. OR ee ONAN 
S 99 8.¢ 820 08 £-G¢ 0&z L¥-T 69-1 OL9: 66L- Ivg. TP Se ee atone 
3S ee SS 
5 64 0-9 096-62 V-CF ZO ore LG-T PLO: 6&L- O19. 6G fs goqureoeqy 
he LL a) 990-08 1-8 +6 eG-1 98-1 99. L8L- 61g. LO erage AON 
< OL Pg 028-62 1-69 281 29-1 ELT Fe: 020-I | L59- PG. [oe ara O10 
3 99 9G 00-08 9-89 18% 9¢-1 6L-1 16); O16: 620. > 15.66) a eereqmtaydeg 
-S 6¢ L-9 920-08 G-L9 99T GGT G9-T 608: P86: PE9: GG = ee easn ony 
rS 6¢ 0-9 190-08 1-69 6FI 9F-T 66-1 198: 620: | #02: S23, Ee ee ae 
% 09 $9 Sr0-08 6-9 991 68-1 GF. 1 O18. 986: ECL, CO Se a ee ee amie 
3 69 8-9 F10-08 1-39 99T CFI 0¢-1 Teh: TL8: 16¢- UG REPS Sear Aw 
es 9¢ 8h 168-6 9-6 $6 89-1 6LT ecg. TL9: PSF. TO. ee er ay 
: 69 Gg G16-66 9-9F ZOE 69-1 69-1 868: O8F- org. OG Sa eee ee OE 
© 69 L-9 602-08 1-4 Lge 09-T SLT SGP: 12%: 9¢8- Be Per meneneeste Arensqeny 
5 9b 6-¢ 666-08 0-1F 00F oF 9¢-T OIF. Q8F- Ore: Og [Sn ee Svenwalp 
oO 7 “soqoul 5 aS oe a oi 
2 /Seime |e] ements | ier) a | ef | ef me | te | 

S 

Si ‘T Fav, 


SE 


an Elster and Geitel Electrical Dissipation Apparatus. 481 


§ 5. The large size of g and q' is due not to the existence 
of any considerable number of exceptionally high individual 
values of g, but to a persistent tendency of a_ to be sub- 
stantially larger than a,. This is readily seen by reference 
to Table II. The first line shows the total number of occasions 
when g lay within the limits specified, the second the 
percentages which these form of the whole 411 individual 
observations. 


TABLE II. 


Q. ea aE te.B: | Bige Ser| Sets 

Number of occasions ...... ie am 61 17 

Percentage of total ......... | TO 71 wD 5H 4 
| 


Of the 42 days when a, exceeded a_ no less than 15 
occurred in December or January. It is a curious fact, 
which may be worth noting, that low values of g seemed to 
have a tendency to occur on successive days. On one 
occasion values less than unity happened on 4, and on three 
occasions on 2 successive days. 

Of the 17 days when g exceeded 3 only two occurred in 
the summer months May to August. 

The largest value of g actually recorded was 8'3Y—in 
November 1907,—but in this and one or two other cases 
where g was exceptionally high or low, some doubt may 
reasonably be felt as to the accuracy of the result. 

At Kew, a large value of g means usually a very small 
value of a,,a small value of g a small value of a_. Now 
individual observations cannot claim any very high pre- 
cision, and when dissipation is small the loss due to defective 
insulation is apt to represent rather a high fraction of the 
total, so that the probable error in a low value of a, or a_ is 
very considerable. 

$6. Table I. shows a well-marked annual variation in 
dissipation, though obviously a good many more years’ 
observations would be required to give smooth results, 
Taking the mean of a, and a_, we have the maximum 
dissipation in June and July and the minimum in January, 
February, and March. The annual variation is the opposite 
of that in P, but the difference between summer and winter 
is less pronounced. It should, however, be remembered that 


Pha, Mag... 6. Vol. 20. Non Mire Sep, 1910.) 2 1K 


———————— 


A482. Dr. ©. Chree: Results obiained at Kew Observatory with 


the results refer to only a small fraction of the day, 
viz., 2 to 4 p.m., and that if we were dealing with mean 
results from the whole 24 hours the annual variation in a, 
and a_ and the relation to P might be different. 

We know that in the case of P the character of the regular 
diurnal variation varies with the season of the year. In 
summer, the element is between 2 and 4 p.m. notably below 
its mean value for the day, but at midwinter at this hour it 
is if anything above the mean. Thus the annual variation of 
P shown by Table I. differs from that shown by the 24-hour 
mean, in the direction of increasing the difference between 
summer and winter. Jn the case of a, and a_ it is at least 
as likely as not that the opposite is true. 

In the case of g and gq’, annual variation if existent 
appears to be small, but whether accidentally or otherwise 
the four equinoctial months all give values above the 
mean. 

§ 7. In order to see whether a, or a_ exhibited any 
parallelism to P or to any meteorological element, the days . 
of each month of the three years were arranged in two 
equal groups, according to the value of the element under 
consideration. When temperature, for instance, was being 
considered, the two groups were composed respectively 
of the warmest and coldest days. If there happened 
to be an odd number of days of observation, the day which 
was central as regards temperature was omitted» Mean 
values were found for each of the groups, for tempera- 
ture, a,, a_, a_/a,, and P, as well as the corresponding 
values of (Sa_/Sa,). Calling these two mean values for 
any element from any one month of the year m and m’, and 
distinguishing the three years by the suffixes 1, 2, 3, means 


M=(m+m,+m3)/3, and Maa +m! + ms!)/3 


were then calculated. The means of the 12 M’s and the 
12 M”s were ascribed to the year as a whole, the means of 
the M’s and M”s for November, December, January, and 
February were ascribed to winter,and so on. The difference 
between the final means M and M' for the year, or for any 
season, was taken, and in the case of a,, a_, and P, this 
difference was expressed as a percentage of the mean value 


of the element for the season. The results thus found are 
given in Tables III. to VI. 


an Elster and Geitel Electrical Dissipation Apparatus. 


483 


TaBLe III.—High Potential ~ Low Potential. 


Differences as percentages of seasonal mean. 


_ Winter 
| 


| 


| Equinox...... | 


Summer...... 


eee eee 


G1. 


a q. | Or 
—21 | +44 | -3 
SPB ik CPE td 
298 +5 +2 
sti +4 8 


Taste [V.—High Temperature ~ Low Temperature. 


Equinox 


Summer 


Equinox 


Summer 


| 


peseee 


TABLE V.—High Pressure ~ Low Pressure. 


Temperature 


Differences as percentages of seasonal mean. 


Difference. 
One, 
; x 
81 +12 
ital +22 
84 +14 
8:0 + 4 


a q: 
+19 + 7 
+27 — 4 
eo Eo 
+3 +2 


Soiled 


Barometric 


! 


Differences as percentages of seasonal mean. 


Pressure 

Difference 
(inches). a4. 
0:337 —25 
0-875 — 2 
0:323 —41 
03138 —28 


2K 


a— qd. 
asa Ba 
= —12 
— 28 +12 
—3l | 0 


LSS) 


484 Dr. CO. Chree: Results obtained at Kew Observatory with 
Taste VI.—Much Cloud ~ Little Cloud. 


Differences as percentages of seasonal mean. 
Cloud Ni Ph eG 
Difference. | 

an; Ge, q. Gi). Pe. 
PGR ncca. 53 +21 +21 —5 +1 —14 
Winter ...... | 6:2 sea ABB 2) ie Bia 0 —19 
Hquinok,..|. 50 | +18 |' 41% —6  -l —7 
Summer...... | 4:7 +14 +18 —1 ; +83 —13 


To elucidate Tables III. to VI., take Table IV. as an 
example, and consider the case of the “year.” The means 
of the 12 monthly means derived from the groups of warmest 
and of coldest days were as follows : 


| Temp.| a+. | We Y- Ga: Pe 
From groups of warmest days ... A94 | “577 | “876! 170) a2) 2s 
| From groups of coldest days ...| 51°3| °513| -726| 1°58| 1-41] 267 
Excess on warmest days ......... +81 |+:064 +:'150|)+012)4+011)/— 38 
Corresponding mean values...... ida se ‘p41 | 799) 163)" a) aaa 
Excess as perceniage of mean .... .... | +12 | +19 | +7 | +8 | —15 


§ 8. Before discussing Tables III. to VI. it is convenient 
to introduce two other Tables which present another aspect 
of the case. The comparison of groups of days, as already 
explained, was made for each of the 36 months of the three 
years. If a close connexion exists between a, or a_ and 
any of the other elements considered, then there ought to be 
a substantial majority of individual months in which the 
connexion appears. This is the aspect of the case that is 
dealt with in Table VII. , 

A value near 18 indicates little or no connexion. <A value 
much above 18 indicates a marked tendency for the two 
quantities compared to be large together, while a value much 
under 18 signifies that the one quantity tends to be large 


an Elster and Geitel Electrical Disstpation Apparatus. 485 


TasLeE VII.—36 months taken individually. 


Number of months when largest 


value of 
l 
Gis.) C q: q' P 
in group of days of highest P_ ............ 21 LO O20 20 — 


temperature.| 24 | 26 19 21 133 


” 7 %) 


pressure...... 10 9 205 | 20 28 


» ”? ” 


of most clouds. 5...:...2: 25 |, 28 15 Lt |, It 


when the other is small. For instance, in 9 months the 
mean value of a, was larger in the group of days of highest 
potential, leaving 27 months in which the larger mean value 
of a, belonged to the group of days of lowest potential. 
The tendency is thus for high values of a, to be associated 
with low values of P, and conversely. A case in which the 
means from the two groups of days were exactly equal was 
counted as 3. 

If, instead of taking the three years separately we take 
them combined, we get the following results instead of those 
in Table VII. The number of months being now 12, the 
criterion for association is a decided departure from the 


figure 6. 
Taste VIII.—3 years combined. 


| | Number of months when largest 


value of 
a a. g iP 
in group of days of highest P ............ 0 1 (= — 
* ; 33 temperature. 84 10 9 3 
99 ¥ a pressure....... 3 3 7s i 
| * - of most cloudy: 24.00: hog 11 5 ye 


§ 9. The somewhat curious fact that the difference between 
the two mean values of P in Table III. was in each season 


exactly the same fraction of the mean seasonal value, makes 


the comparison especially instructive. 
It is clear from Table VIII. that there is a distinct 


| 
tl 


486 Dr. C.Chree: Results obtained at Kew Observatory with 


association, the whole year round, of high values of a, and 
a_ with low values of P, but this association according to 
Table III. is decidedly less marked in summer than in winter 
or equinox. There is, however, room for doubt whether the 
apparent difference between summer and the other seasons 
is real. It is due almost entirely to the one summer 1907. 
In all four months of that season, the group of days of 
largest P had the higher mean value of a_, and in two cases 
out of four the higher mean value of a,. 

Tables VII. and VIII. both suggest that a_/a, has a 
slight tendency to be large when P is high, but the 
numerical differences for g and g' in Table III. are too small 
to rely on and tend to differ in sign. 

Tables IV., VII., and VIII. agree in associating large values 
of both a, and a_ with high temperature, but the apparent 
closeness of the association is widely different at the different 
seasons. It is conspicuous in winter, but tends to disappear 
in summer. In fact, when the three years are combined, 
the two hottest months July and August associate high 
values of a, and a_, not with the higher but with the lower 
temperature group of days. Of the 24 individual months in 
Table VII. which associate high values of a, with high 
temperature, no less than 11 are contributed by winter, so 
that the two other seasons only contribute 13 out of a 
possible of 24. The association of high values of a_/a, with 
high temperature is, according to Table IV., pronounced in 
equinox but not in the other seasons. 

The relationship between temperature and potential pre- 
sents closely similar features in its annual variation. 1t is 
marked in winter in Table [V., high temperature going with 
low potential, but not in equinox orsummer. As Table VII. 
shows, high temperature was associated with low potential 
in 224 of the 36 months, but no less than 11 of these were 
winter months, and an actual majority of summer months 
associated high temperature and high potential. 

The apparent relation between dissipation and barometric 
pressure is in several respects the exact opposite of that 
with temperature. According to Table V., high pressure is 
at all seasons associated with low values of a, and a_, but 
the association is much less apparent in winter than in 
summer. This fact is all the more striking because the 
mean pressure difference between the groups of days of high 
and low barometer was conspicuously large in winter. As 
shown in Table VII., the number of individual months which 
associate high values of a, and a_ with high pressure is 
appreciable, but winter is responsible for 6 out of 10 in the 


an Elster and Geitel Electrical Dissipation Apparatus. 487 


case of a,, and for 5 outot 9 in the case of a_. In only one 
out of 12 summer months was there an association of high 
values of a, and a_ with high pressure. Tables VII. and 
VIII. agree in indicating a tendency for high values of 
a_/a, to be associated with high,pressure ; but Table V. gives 
no support to this conclusion except in equinox. The 
association of high potential with high pressure is marked. 
According to Table V. it is reduced in winter, but still 
winter contributes 9 months out of the 28 in which the 
associationship appears. 

Tables VI., VII., and VIII. agree in associating high 
values of a, and a_ and low values of P with the prevalence 
of much cloud. The apparent influence of cloud seems 
remarkably alike in a, and a_ according to Table VI. It 
appears greatest in winter, and is distinctly larger in the 
case of dissipation than in that of potential. 

_§10. The question of the influence of cloud is complicated 

by the facts that there are a number of different types, 
representing different meteorological conditions, and that 
the relative frequency of the various types varies with the 
season of the year. An attempt was made to ascertain 
whether the electrical phenomena associated with the different 
types of cloud differed. The difficulty at once presented 
itself that upper and lower clouds are usually both present, 
and that not infrequently there is more than one type repre- 
sented, both in the lower and in the upper. There were, 
however, some types which occurred alone in a sufficient 
number of instances to warrant the hope that conclusions of 
fair reliability might be obtained. These were stratus 
Gneluding ordinary low stratus and alto-stratus), cumulus 
(at all levels including fracto-cumulus, but not  strato- 
cumulus), and cirro-stratus (at high levels). In a good 
many months there were no representatives of one or other 
of these types, and in many other months there were only 
one or two representatives. Thus the method of deriving 
mean values of dissipation and potential for days of each 
species of cloud in éach month was unworkable. The plan 
adopted was to examine each day by itself, and classify 
separately the corresponding a_,a,, and P, according as the 
value was above or below the mean value for the month. 

A difficulty, however, presented itself in the interpretation 
of the results. None of the three quantities a_, a,, or P 
has its values occurring symmetrically with respect to its 
arithmetic mean. Taking all the observations, the number 
of occasions having values larger than the mean is very 
decidedly less than the number having values below the mean. 


nt 


488 Dr. C. Chree: Results obtained at Kew Observatory with 


Thus no inference can be drawn from the results for any 
particular type of cloud without considering the corr esponding 
results from all days of observation. This comparison is 
made in Table IX. It gives the number of days of each 
class, the corresponding mean amount of cloud and the per- 
centage of occasions in which the value of a 45 4_, or P was 
above the average derived from all days of observation in the 


month to which the observation belonged. The results are 


given to the nearest 0°5 per cent. 


TABLE LX. 


| Mean Percentage of cases when 
Mh hi al Number) cunt | Value above the monthly mean. 
y pe O Gay. | of days. a = bes Len 
’™ | of cloud.| 
| | a1. OHS P. 
Vs Wee aera by OS tiara 
HUNG ne ee ar V2 BRI gate Co Mi se 44°5 42 
Days of Stratus ...... 50 90 | 47 41) capo 
| 
Dee Oreurethyht eae BR ul gare Hill dl 
,,  Cirro-stratus. 85 54 46 | 939 40 


Tn the cases of cumulus and cirro-stratus the mean amount 
of cloud is less than the mean from all days, but the difference 
is not large, so that any conspicuous pecniarity in the results 
for either class, if not “accidental,” is presumably really 
dependent on the type of cloud. We thus infer that the 
presence of cumulus has a distinct tendency to be associated 
with low values of P and of a,, but with high values of a_ 
and so of g. The presence of cirro-stratus seems to have 
exactly the opposite effect so far as dissipation is concerned, 
but the apparent association with high values of a, is at least 
doubtful. The apparent depression in P is also too small to 
possess much significance. Stratus appears to be associated 
with a slight rise in a, and fall in a_, and a very decided 
fallin P. The amount of sky covered on days of stratus is, 
however, much larger than on an average day, so that the 
apparent effects on P and a, may be due to the quantity of 
cloud and not to its type. The examples of stratus were 
mostly from the winter months, and so from the season when 
Table VI. makes the influence of cloud on P largest. 

§ 11. Amongst the other meteorological conditions con- 
sidered were right sunshine (as opposed to faint or no sun- 


shine), clearness of the atmosphere (as opposed to haze or 


an Elster and Geitel Electrical Dissipation Apparatus. 489 


mist), and the limiting form of extreme clearness distin- 
guished by the letter v (high visibility) in the meteorological 
records. Mean values of a,, a &c. were got out for the 
days of bright sunshine in each month, and the algebraical 
residues remaining after subtracting from these the corre- 
sponding means from all the observations of the month were 
totalled for the 36 months. The mean of this sum of differ- 
ences was finally expressed as a percentage of the mean value 
of the element for the three years. These percentages for 
the several elements appear in Table X. The same method 
was applied in the cases of clear atmosphere and of high 
visibility. 


TABLE X. 
— 2 
Percentage excess of element on 

| 'Number| representative day of special type. 
: | of days. 
a. a. q. E. 

Bright sunshine ......... pani Vo 4 —15 0 + 6 

: | 

Clear atmosphere .. ... | 144 Sibi A) a ie Baa tlerenns L | —10 
| High visibility ......... Ee rN BL td DLL = 


To illustrate the interpretation of Table X., take the case 
of bright sunshine. Out of the whole 411 days of observation 
there were 125 on which there was bright sunshine during 
at least the greater part of the dissipation experiment. On 
the average day of bright sunshine a, was 14 and a_ 15 per 
cent. below its mean value for the season, while P was 6 per 
cent. in excess of its mean value. There was no appreciable 
effect on g. The days of clear atmosphere include the 24 of 
high visibility. They exhibit the exact opposite of the 
phenomena exhibited by the days of bright sunshine, while 
the days of high visibility are specially conspicuous for the 
large values of the- dissipation. Whether the apparent 
reduction of g on days of high visibility may not be in part 
at least ‘accidental’? is open to some doubt, on account of 
the comparative fewness of the days. High visibility was 
mainly confined to summer months, no single example being 
encountered from November to March. 

The difference apparent between days of bright sunshine 
and days of clear atinosphere may appear at first sight im- 
probable, as one is apt to regard the two meteorological 
conditions as naturally coexistent. This, however, is by no 


4s 


i\t 
A Sh 


490 Dr. C. Chree: Results obtained at Kew Observatory with 


means the case at Kew, more especially in the afternoon, 
when haze is a frequent accompaniment of sunshine. 

§ 12. It was originally intended to treat relative humidity 
in the same way as temperature, pressure, and cloud, and this 
was actually done for one whole year. The mean electric 
results, however, from the groups of days of high and of low 
relative humidity were so closely alike for all the seasons that 
it appeared unnecessary to proceed further. 

§ 13. The last meteorological element to be considered in 
detail was wind direction. In some months one direction 
was so dominant that grouping of days presented difficulties. 
Supposing the wind to be westerly on 9 days out of 11, little 
significance can be assigned to the numerical size of the 
difference between means derived from the 9 days and from 
the 11. Accident is likely to play too large a part. The 
method actually adopted was as follows: the days of each 
month were grouped under the four fundamental directions, 
N, E,S,and W. Ifthe wind were N.W., or N.N.W., or 
W.N.W., it was grouped under both N and W. The values 
of a,,a_, g and P were then found for each of the four 
directions—or for all that were represented—for each month. 
Taking for example a,, the direction giving the largest mean 
value was regarded as taking the first place in a competition, 
the direction giving the next largest mean value the second 
place, and so on. Out of the whole 36 months there were 
19 on which the largest mean value of a, was associated with 
a South wind, as compared with 10, 4, and 3 respectively on 
which the largest value was associated with a West, a North, 
and an Hast wind. The second place on the list was taken 
7 times by a South, 19 times by a West, 6 times by a North, 
and 4 times by an East wind. These results form the first 
two rows in Table XI. under a,. The data for a_, g, and P 
possess the same significance. 

The sum of the figures for “first” and “second” always 
totals up to 36, but the same is not true of the figures for 
“third” and “fourth” because in some months not more 
than 3 or even than 2 wind directions were represented. 

A glance at Table XI. shows a very pronounced association 
of high values of a, and a_ with southerly and westerly 
winds, and an equally pronounced association of high values 
of P with northerly and easterly winds. gq seems to be inde- 
pendent of the wind direction. 

The influence of wind direction on a,, a_, and P is much 
less marked in winter than in the other seasons. Winter 
contained 5 of the 7 months in which a, was highest with a 
N. or E. wind, 7 of the 9 months in which a_ was highest 


an Elster and Geitel Electrical Dissipation Apparatus. 491 
TABLE X1I.—Influence of Wind Direction. 


Times first ...) 4 


Wind Direction.| N. | E.| 8. |W.| N.| E.| 8. |W.) N./ BE. | 8. |W.) NL] E. 


second...| 6 4 Pi [a 6| 4 
mimes 16 }10 | 316.) 15} 10 
meee toe | ia |b} LP EY ) 11 | 


a+. a—. | q. P. 


(sy) 
— 
oO 
je 
(am) 
pay 
or 


with a N. or E. wind, and 3 of the 4 months in which P was 
highest with a 8S. or W. wind. Confining ourselves to firsts 
and seconds in Table XI., and combining Kast with North 
and West with South, we obtain the following figures for 
Winter (i.e. 12 months of the 36) :— 


North and East. | South and West. 


i : cee aay: 7 
Nimiber OF. chee.) a. bey Gera ale Ore « deg 
Reprise” (bss. .30se bay F 9 ri 2.15) 
|’ SecopdS......<z «+. 4 4 v4 8 | 8 5) 

H | | 


In the case of dissipation South and West have a distinct 
majority of seconds, but absolutely no majority of firsts. 

The influence of wind direction is a complicated question. 
Kew Observatory is situated in a large park, which is bounded 
on the east by the extensive Kew Pordens and to the north- 
east across the Thames lies Syon Park. No inhabited 
buildings are at all near in the direction from 8.H. to N., 
while Isleworth approaches to within 3 of a mile across the 
river in the N.W. direction. St. Margarets, the next suburb 
up the river, follows immediately on Isleworth, and extends 
to Twickenham. Richmond is about 3 of a mile away in 
the direction from 8. to 8.E., and behind Richmond lie 
Richmond Park and Sheen Common. ‘Thus there is a com- 
paratively narrow fringe of houses—mainly dwelling-houses— 
extending round from N.N.W. through W. to 8.E. within a 
mile radius, while from N.N.E. to E. there occurs the great 
mass of London, but few houses come inside a 14 mile radius. 
Sometimes when the wind is easterly the curtailment of day- 
light by London smoke is very apparent, but this is mainly 
in the winter months when the influence of wind direction, 


492  Llster and Geitel Electrical Dissipation Apparatus. 


as we have just seen, is least decisive. If London smoke is 
the really effective influence, then the effect should be con- 
siderably dependent on the hour of the day, and when most 
marked at Kew should be least marked at a station to the 
east of London. <A comparison of simultaneous results from 
Kew and such a station should be decisive. If the low value 
of the dissipation and the high value of the potential gradient 
at Kew arise from its proximity to London, then the electrical 
conditions in the heart of London itself are presumably highly 
abnormal, and it seems unlikely that great abnormality in 
any atmospheric condition will be without some influence on 
living ea exposed to it. 

§ 14. B. Zolss has described a remarkable parallelism 
between dissipation and the size of the daily range of 
declination at Kremsmunster, and has explained this as a case 
of cause and effect, dissipation representing a vertical electric 
current, and declination change the consequent effect on the 
magnetic needle. If such a connexion could be definitely 
established, it would be a result of great physical importance. 
hod conclusions are given without criticism in Mache and 

y. Schweidler’s textbook of Atmospheric Electricity, and it 
fee appeared worth while seeing whether any confirmation 
was derivable from Kew results. One point deserving 
attention is that the dissipation observations at ‘Kew covered 
only two hours, while the declination range represents mag- 
netic changes during a larger fraction of the day. 

The diurnal range 2 of declination, as Zolss himself recog- 
nized, agrees with ‘dissipation i in being considerably larger in 
summer than in winter. If the two phenomena are compared 
in a way which does not eliminate annual variation, then a 
conclusion similar to Zélss’ is practically certain to be reached. 
But when the effects of annual variation are eliminated by 
subdividing the observations of each month into two equal 
groups composed respectively of the days of largest and of 
least declination range, the Kew results at least afford no 
support to Zélss’ conclusions. It appeared sufficient to con- 


sider one year, 1908. The final means derived from the 


12 pairs of monthly groups were as follows :— 


Mean value of 
Groups. S ae l f 
eclination 
Range. PCa +a_): Fr. 
| | 
| Days of largest declination range ... 17°6 | 549 210 | 
| | 
zs least s fe 9'-2 | ‘581 [2 Gy 
| | 


SE a ee 


Stability of Superposed Streams of Viscous Liquids. 493 


The example of Zélss has been followed in taking the mean 
of a, and a_ as a measure of the dissipation. 

The difference between the groups as regards declination 
range is most substantial, but the difference between the 
corresponding values of dissipation and of potential is too 
small to possess any certain significance, and, as it so happens, 
the smaller value of the dissipation appears in association with 
the larger value of the declination range. 

As already explained, summaries of results obtained with 
the Elster and Geitel apparatus have been published for a 
number of stations abroad, and in most cases investigations 
have been made as to the apparent association with different 
meteorological conditions. For information on these points 
the reader is referred to A. Gockel’s “‘ Die Luftelektrizitit”’ 
and to Mache and v. Schweidler’s ‘* Atmosphirische 
EHlektrizitat.”’ | via 


XLIX. On the Stability of Superposed Streams of Viscous 


Liquids. By W. J. Harrison, B.A., Fellow of Clare 
College, Cambridge*. 


§1. E this paper it is shown that, if two streams of viscous 

liquids are moving uniformly in laminar motion, 
one of which is superposed on the other, and both are of 
great depth, the motion will be unstable under certain 
circumstances for disturbances of the interface which are 
of greater wave-length than some determinate limit +. It is 
clear that if instability ensues in any particular case it will 
be for great wave-lengths, comparatively speaking, and 
not for small ones, since, in the latter case, the motion is 
equivalent to two streams flowing with the same uniform 
velocity, as far as the disturbance is concerned. Lord 
Rayleigh has found a similar result when treating the 
disturbances between two streams of a liquid moving in 
opposite directions with uniform velocity, but separated by a 
transition layer of liquid in which the velocity changes 
uniformly. He says, “It appears, therefore, that so far 
from instability increasing indefinitely with diminishing 
wave-length, as when the transition is sudden, a diminution 
of wave-length below a certain value entails an instability 


* Communicated by the Author. 

+ It ought to be clearly stated that the stability here discussed is only 
for the case of particular modes of disturbance, namely, those originating 
from a disturbance of the interface. The arguments of Osborne Reynolds 
would seem to show that the motion must be unstable for a general 
disturbance, as there are no lateral boundaries to determine a limit to 
the instability. 


494 Mr. W. J. Harrison on the Stability of 


which gradually decreases, and is finally exchanged for 
actual stability ”’ *. 

I have given below three methods of solving the equations 
of motion, of which only one has been employed. The 
other two are too cumbrous for practical use, though more 
rigorous. There still remains another method, which has 
the advantage of not needing the same assumptions for the 
purposes of approximation, and which is especially adapted 
for very viscous liquids, but can only be employed for 
disturbances of great wave-length. However, as the stability 
of the motion is determined by its stability for great wave- 
lengths, this method will furnish the precise information we 
need. I hope to develop this solution in a future paper. 

§2. We shall confine the problem to two dimensions and 
take the origin of coordinates (#, y) in the undisturbed 
interface, the axis of 2 being in the direction of flow, the 
axis of y vertically upwards. 

The equation which is satisfied by the stream function W 
can easily be shown to be 


ey’ — Sve =(F 2 - She atl 


where v is the kinematical coefficient of viscosity. 
We shall take the undisturbed motion of the lower liquid 
to be given by 


Wy = By + $Cy’. 


This implies an infinite velocity at y =—o, but as we are 
only concerned with the condition near the interface this 
need not cause any serious trouble. Lord Rayleigh makes 
the same assumption in his work on motion past a corrugated 


wall Tf. 
For the upper liquid 


ho = Bly + 302’. 
where we must have 
B= 6, 
yp =v'p'C’, 


by reason of the continuity of velocity and traction across 
the interface. 


* Scientific Papers, vol. i. p. 480; Proc. Lond. Math. Soc., xi. p. 68, 
1880. 


+ Scientific Papers, vol. iv. p. 89; Phil. Mag. xxxvi. p. 368 (1898). 


Superposed Streams of Viscous Liquids. 495 
The complete solution will be given by 
VHaW+y, WH +, 


where wW, y’ are due to small disturbances, whose squares 


are to be neglected. 
We assume vf to be of the form 


ap = E(y eit, 
On substitution for VY in (1), we obtain 


AS — 2) F(y) a (« + kB + iy) S _ 2B) ==) 03ii¢2,) 


a ale! 
Writing 


oi : 
5B )E@) = fw). 
we can put (2) into the form | 
0” 
(5,3 as 2) Ay) = (a+ ehB + okCy) fly) =0. . (3) 


§3. Before proceeding further with the solution of the 
problem, I wish to insert here the solution giving the form 
of the free-surface of a stream of uniform depth flowing 
over a corrugated bed, over which it is assumed that the 
jiquid can flow without experiencing any resistance. 

In equation (2) we have C=0, a=0 

It is easily shown that 


F(y) = aye™ + bye + age + dye-™, 
where 
? = kh? +ikB/v. 
We may take the bed to be 
y =—h+Boe™, 
and the free-surface — 
i be, 
Writing down the usual conditions 
v= —uyh, Pay = 0 at yY= —h+ Boe™, 
We Oe Pop = 0, Pon = CONS aly ==) Be™, 


and eliminating the constants, we obtain the relation 


i 


SSS ESS SS SSeS = 
= = SSS = 


496 Mr. W. J. Harrison on the Stability of 
between 8 and {, in the form 


| P cosech kh—Q cosech Ah | Bo 
=| P coth kh — Q coth rAh+ gk?/v |, 
where P = ky(2kh? +ckB/v)?, 
Q = 4h 


This leads to a finite expression for 8 for all velocities of 
the. stream, and also gives the difference of phase between 
the corrugation and the wave- -profile. When the wave- 
length is small compared with the depth the solution differs 
w idely from that in the case of a non-viscous liquid*. 
When the stream velocity is great the amplitudes in the 
viscous and the non-viscous cases are equal. 

§4. There are two ways of rigorously solving the equation 
(3). The first is that which was given by Lord Kelvin in 
his paper on ‘ Rectilineal Motion of Viscous Fluid between 
two Parallel Planes”? +. But in order to adapt this method 
of solution to our problem it is necessary to employ very 
complicated integration of Fourier’s type. 

The second rigorous solution can be obtained in terms of 
the Bessel’s functions Ji, I:; this problem thus affords a 
second example of the use of these functions in physical 
analysis. The double integration involved in the solution of 
equation (2) from that of (3) can be expressed, first of all, 
as a triple integral and then reduced to a single integral by 
the aid of Dr. J. W. Nicholson’s results in his paper ‘‘ On the 
Relation of Airy’s Integral to the Bessel’s Functions” f. 

But again the analysis involved in the use of these 
functions is too complicated, and we are forced to consider 
an approximate solution §. 

§5. Returning to equation (2) we shall solve by successive 
approximation on the supposition that itC is small compared 
with «+7kB. To satisfy this assumption it is not necessary 
in general that C sbould be small. 


« Cf. Lamb, Hydrodynamics, 3rd edition, p. 389. 

+ Phil. Mag. xxiv. p, 192 (1887). 

t Phil. Mag. (6) xviii. pp. 6-17 (1908). 

§ Since writing this my attention has been drawn to two papers by 
Prof. W. M‘F. Orr in the ‘ Proceedings’ of the Royal Irish Academy, 
1907, in which he uses these functions in the similar but less complicated 
problems of the stability of motion of a single liquid flowing between 
parallel planes. 


Superposed Streams of Viscous Liquids. 497 


The solution suitable for the lower liquid is to a first 
approximation 


F(y) = Lely + Ne = foly), 
where WM = kh? +(a+ckB)/v, 


and the real part of » is to be taken positive. 
To a second approximation 


Fy) = folly) thy), 


where we have 
ay --*) Aly)-—O?— “H(% —i2) Ay) ="Lyoe—B)New, 
The particular integral is 
Fiy) = Byte + Que, 
where P =2kC. N/4ay, 


ga WP) +4? k0 
me Mr2—k2) * Ady 


To a third approximation 
Jaly) = ty? + Sy? + Ty? + Wye, 
where R = tkC P/8ry, &e. 


—— N. 


Thus for the lower liquid we assume 
B(y) = Let’ + New[1+ Ny + Noy? + Noy? + Nay], 


where N, = (Q4+ W)/N, 
N, =(P+D/N, 
No =p) a. 
N, = R/N, 


and the stream function becomes 
VY = By+4Cy?+ F(yje**, 
For the upper liquid 
B’(y) = Lie“ + Nle-*4| 14 Nyy + Noy? +No/y? + Nyy], 
and YW = By+3Cly?+ F(yjem te, 
Phil. Mag. 8. 6. Vol. 20. No. 117. Sept. 1910. 2 L 


7 


498 Mr. W. J. Harrison on the Stability of 
§6. The conditions at the interface are 
re se 


=A 
ie af 0 
we vt ’ 
eel ; 


where pz, Py are the tangential and the normal tractions 
at the interface. 
From the first three we have 


L+N=U4+N) . on). 3s 
kL+(A+N,)N =—kL'+(—-V4Ny)N',. . (5) 
pv 2h L+N(k? +A? + 2AN;+2N,) | 
= p'v'[ 2k? L’ + N'(k? +22—20'N,’+2N,’)] . (6) 
From the two equations of motion 


Ou ou Ow 1 Op 
rT hs ar pee ee 


Ov wom Ane Lied OP cee 
at Oe Oy. poy oe 


we find 


ee) = — gikn—(a+ikB)(kL+AN +NN,) +ikC(L+N) 
+v[A(A2—k?) + 3X2N, —/°N, + 6AN,+6N, |, 
where 7 is the elevation of the interface, 
n = —tk(L+N)/(a+ekB) 
= —ik(L/ +N’')/(a-+ckB). 


Be fost 


=—(*) — Qvik(kL+AN +NN)). 
p/o 


eS 


Superposed Streams of Viscous Liquids. 499 
Thus we have as our final condition 
plak?(L+N)/(2+ckB) +(2+ckB)(KL+AN + N,N) —2kC(L 4+ N) 
—vN {r(A? — k?) + (3A? —27)N, + 6AN, + 6N33} 
+ 2vk{kL+(A+N,)N}] =p'[g?(L'+N’)/(a+ckB) 
+(a+ikB)(—kL'—v'N'+ NN’) —ckC'(L' +N’) —V N44 — (0? = Fe?) 
+ (3X? —k?)N,/—6AN,' +6N,) 5 +20 P{RL/4+(—r'+ NN 3] . (7) 


Now we can write the equations (4), (5), (6), (7) in the 
form 


L+N=L1/4+N%, 
kL+n,N =—kl/—n'N’, 
2h pvL +ngN = 2k*p'v'L'+n,’N’, 
LL+tn3N = 1,'L/+n,/N’. 
The eliminant of L,N,L,/N’ is the required period equation, 


k ny k ny’ = 
2hov ng = —2k*p'v’ — — nn’ | 

l, ng —l’ —ns; 

1 1 -1 —1 


§7. We can approximate to the solution of this equation 
on the supposition that v and v are both small. Now it may 
easily be shown that the term of highest order in 7, is of 
the order v—3, those in /,, ms, n3 are of the order 1. Hence 
retaining the terms of the two highest orders only in the 
period equation, it reduces to 

(1, —1!)(myng' + ngny’) + kng( —1, —L/ + 273’) 
+ kng! (1; + L,/ —2n3) = 0. 
The first approximation is given by 
“ty — e — 0. 
Now to our order of approximation 
L, = (gk?/a 9+ ka, — ukC)p, 
L,! = (gk? /ay—kay —tkC')p'. 
(p—p!)gh? + (p+ p!)hau? — uk(Cp —C’p')ao = 0, 


Hence 


where a has been written instead of «a+1kB. 
2 [,2 


ad = 


500 Mr. W. J. Harrison on the Stability of 


This equation leads to the solution 
ag= +2] {(Cp — Cp!) +4gk(p? —p)}? + (Cp—O'p’) ]/2(p +p") 
= B(say), 
like signs being taken together. 
There are two modes propagated in opposite directions 
with different velocities relatively to the stream velocity at 
the interface. If Co=C'p', we obtain the usual result. 


For the purposes of the next approximation we have to 
substitute 


Ny = (a/v)?— 5ikC/4eo, 
he (ao/v')2 + 5ikC’/4a, 
Ng = P%o a 5ikCp/2(agv)?, 
nq’ = p'ag+ StkC'p'/2(aov')2, 
ng = gk? p/p, 

ng’ = gk'p'/ao. 

In these approximate values the quantities R, 8, T, W 
are not sufficiently important to contribute any term, and 
therefore might have been omitted. But this would have 
given the solution the appearance of being correct to the 
first power of 2kC/(a#+ckB) only, instead of to the second, 
as we require it to be. 


To a second approximation « = @++, where it is quite 
easily shown that 


fe ites 27 Qe —(N) 18 
y= ok (dnraiy He (p pw) [S?+ (Ce— C'p') | 
(p+p)(pvv +p'vy) 
Aghk(o + p') + (Cp—C'p')(C +0) + (C+ CDS? 
Agk(p? — p'?) + (Cp—C'p’)? + (Cp—O'p)8# ” 
where | S = (Cp - O'p')? + 49k(p?— p”). 
With the exception of the first choice of signs, like signs 
are to be taken together. 
This result can be verified by putting C=C’=0. It then 
agrees with the former result * for two superposed fluids at 


rest, provided we take the negative sign in the first choice. 
The sign of the real part depends upon that of 


~ Agk(p?—p’”) + (Cp — C’p’)? + (Cp—C'p') 82 * 
* Proc. Lond. Math. Soe., ser. 2, vol. vi. p. 899 (1908). 


Superposed Streams of Viscous Liquids. 501 
(a) Cp>C'p’. 


Now when £ is very small 


4gk(p +p’) tree i 
era ere 


= 4gk(p + p'){2(Cp—C'p') —(p—p')(C+C) }/(C +0) 
= 4gk(p + p')(C—C)/(C+C). 


Thus the numerator can be negative when C<(’, but the 
denominator can never be negative. Hence the motion is 
unstable for disturbances of great wave-length when C<C’. 
Remembering that Cp>C’p’, and that vpC=v'p’C’, these 
conditions are equivalent to v’p’<vp, and v’>v. These two 
inequalities are not inconsistent with p>p’, and hence under 
these circumstances the interface will be unstable for waves 
propagated in one direction of length greater than some 
limit. ; 


(6) Cp<C'p’. 


As before, the denominator is always positive. The 
numerator corresponding to one mode will always be nega- 
tive when £ is small, and in consequence the interface will 
be unstable. Since vpC=v'p’C’, we have shown that, if 
y'<y, the motion is unstable for waves of great wave-length. 

Thus putting the two cases together we have shown that, 


if v'p’<vp, v'>y, 
or if vy <y, 


the motion will be unstable for great wave-lengths. Taking 
into account the fact that p'’<p, these two cases are both 
ineluded in the inequality v‘p’< vp. 

There still remains the question of the validity of the 
assumption that 7k is small in comparison with «+ kB, 
or a. Now from the expression for a it is quite evident 
that this is satisfied, except when & is very large, without 
the necessity of assuming that C is small, if the c. G.s. system 
of units be used. 

It is to be noticed that (2+72kB) und ¢kU are not of the 
same dimensions, and therefore the assumption that 2kC is 
small compared with (#+72kB) may not lead to a solution valid 
when y is great. But this does not affect the form of the 
solution when y is small, which is all we require to know for 
the present purpose. 


Clare College, Cambridge. 


fe 02K 


L. On the Nature of the Transition Layer between Two 
Adjacent Phases. By Wm. ©. McC. Lewis, M.A., D.Sc., 
Physical Chemistry Laboratory, University College, London ™. 


T is well known that the “ internal pressure ”’ or “‘ molecular 
pressure ”’ in a liquid is included in the van der Waals’ 


e e a e e 
equation as the correction term 2 Obviously the numerical 


values obtained for the internal pressure by the calculation 
of a and v will be average values obtaining throughout the 
bulk of the liquid. Let us denote such values by K,,. For 
the particular case of water at 0° C. van der Waals himself 
has calculated K,, to be 10500-10700 atmospheres tf. There 
is another method, however, first proposed by Dupré (Théorie 
Mécanique de la Chaleur, Paris, 1869), viz., that K is the 
work required to remove unit volume from the surface layer 
in the form of very thin lamine, and carry them outside the 
range of their mutual attraction. In other words, K is the 
internal work required to vaporize unit volume of the liquid 
at the given temperature. Lord Rayleigh in his work on 
“The Theory of Surface Forces” (Phil. Mag. xxx. p. 285, 
1890 ; Scientific Papers, vol. iii. p. 396), states that “this 
view appears to be substantially sound.” Assuming that the 
volume of one gram of water is approximately the same in 
the bulk of the liquid and in the surface layer one finds that 
for water at 0° C. the value for the internal pressure comes 
out to be about 25,000 atmospheres, 7. e., there is a large 
discrepancy between K,, and Kpupre. Other substances show 
the same peculiarity, viz. :— 


Witwer 2.0.0. K,, = 1300-1430 atm. Kpupre = 2426 atm. 
Ethyl alcohol...... K,,== 2100-2400. 4, Kajupe =F 2a 
Carbon disulphide Kmn=2890-2900 ,, Koupre=4704 ,, 


The differences are so large that they can hardly be 
regarded as accidental, so that one is forced to the conclusion 
that the internal pressure in the surface (call it K,) is con- 
siderably greater than the average bulk value K,. This 


* Communicated by the Author. 

+ An attempt at calculating how Ky» varies with temperature is ren- 
dered very difficult owing to the fact that van der Waals’ “a” is anything 
but constant. Thus, taking as our units atmosphere, litre, grammole, 
for pressure, volume, and mass respectively, the value of a at 0°C. is 
3°467; at 100° C. a = 3:29 (according to Traube), and at the critical 
temp. 362°-4 C. a=5°77 as calculated from the critical data. The 
inconstancy of “6” is even more marked, but does not concern us here. 


Transition Layer between Two Adjacent Phases. 503 


further suggests that the average density in the surface layer 
(call it os gram/c.c.) is larger than o,,, the bulk value, though 
this conclusion really widens the difference between the two 
sets of values. As a matter of fact, as will be shown later, 
the values of K; at ordinary temperature for water appears 
to be of the order of 50,000 atmospheres. 

Wecan see the connexion between Kn, Ks, om, o,most clearly 
by making use of the Laplace expression for molecular 
attraction. The general expression developed by Laplace for 
the internal pressure is (employing the usual symbols) 


(c is the range of molecular action). 
For the bulk of the liquid, therefore, 


Ra o2( (2) dz. 
«0 


For the surface layer 


and therefore, 


Os = om a e ° e e e ° (1) “ 


This expression is, however, not much use to us as it stands, 
since it contains two unknowns, o, and K,. For the same 
reason the value of o, cannot be obtained from what might 
be called a corrected form of the Dupré relationship. Thus 


. 1 s 
o; 18 of course a where vs is the average volume of 1 gram 
Ss e 


of the substance in the surface layer (the corresponding 
quantities in the bulk of the liquid being c,, and y,,). It is 
usual to call vy, or vs *‘specific volumes.” Further, if 
X; represents the internal latent heat of vaporization of 
1 gram, then Dupré’s relationship is 


Ni 
Ka =—=——. ° . . 2 . e . e (2) 
Vs 
To determine vy, or « we must have recourse to Bakker’s 
-relationship—which also requires a slight modification if we 
are to regard vs and vy» as differing in numerical value. 


= 


: 


| 

| 
yl 
i 
| 
H\ 
i 


504 Dr. W. C. McC. Lewis on the Nature of the 


Bakker’s Relationship. 


Bakker *, in 1888, was the first to point out that the latent 
heat of vaporization is given by the expression 


Veas 
A= ( Kadv + p (Vgas— Va), 


eV” 


where vq denotes the specific volume of the liquid. In 
view of the considerations advanced above one can no longer 
speak of magnitudes as pertaining to the liquid state only, 
but must further specify whether bulk or surface. Hvidently 
here we are dealing with surface quantities, so that with 
corresponding notation 


Voas 
rn =| K,dv + p(Veas— Vs) 


s 


(x is the latent heat, external.and internal, per gram). 
Further, van der Waals’ equation gives us for the surface 
layer 
a 


Q 27 
8 


K, = 


= 


so that on integration the above expression becomes 


V. 


1 1 
A= a(> arn ) + P (Vgas— Vs); 
Vs gas 


or approximately, 


Got a a 
eee eee OT Ne re 
» nici 5 Vs 


(3) 
where M is the molecular weight of the liquid in the state 
of vapour. 

The connexion between Bakker’s relationship and Dupré’s 
expression is obvious. 

By means of equation (3) we can calculate v,; knowing 
rX; and a. This has been done for a number of chemical 
compounds given in the following table, which is partly taken 
from a list compiled by J. Traube T. The values refer to the 
boiling-point of the respective compounds. 


* G. Bakker, Dissertation, Schiedam, 1888. 
+ J. Traube, Ann. d. Physik [4] xiii. p. 300 (1902). 


er) 
—) 
“wD 


Layer between Two Adjacent Phases. 


ton 


t 


Tanst 


7 


di 


Substance. 


MGT ORE Y= Sreritax.-concet 
Isopentane ............ 
n Hexane ...... 
n Heptane ... 
eC a a 
EMG Birt « oveVh ess cxw vaveaee 
Carbon tetrachloride . 
Zine chloride............ 
IBGRZONG vessce., hr crest 
Fluor-benzene ....... 
Chloro-benzene....... . 
Methyl formiate ...... 
Ethyl acetate ......... 
Methyl propionate ... 
Methyl alcohol ......... 
A COLE HOMO checker utes ct 
VV PU eh. ovate aees Soh 


* ax is the value of @ at the critical temperature. These values are inserted for comparison with a, The remaining data are calculated from az, 


Molecular 
weight in 
grams. 


114°2 
741 
1538 
260°3 
78°04 
96:06 
112°45 
60:0 
88'1 
88'1 
32:0 
60:0 
18:0 


{ae Ny 
220. |Cals. per 
gram. 
360°0 62:0 
28°0 83°2 
69°0 79°4 
98°4 74:0 
125'8 70°84 
34:8 845 
76:2 46°35 
1125 30°53 
80°25. | 93:45 
85:1 791 
182°0 72:0 
32°9 | 116'1 
75°9 86°7 
80°0 84:2 
64:5 | 267:48 
119-2 89'8 


100 0 |, oan 


Tasie I. 


Ni. 
_— 
Cals. Litre 
per atmos, 
gram, |per gram 
55'3 2'322 
74:9 3146 
72:2 3'0382 
66°5 2°793 
63'8 2'679 
76°2 1814 
418 | 1°756 
27'6 1°159 
84:4 3°544 
71°3 2°994 
64'8 2°721 
105'°9 4876 
73:8 3°309 
762  3:200 
246°8 10°360 
W114 4677 
495°2 . 20°800 


uy 


————~ litre? atmos. 
unit of mass 


= gram 


mol, 


8°68 
11°23 
15°37 
18°69 
22:17 
10 56 
12:04 
16:05 
11:13 
ott 
1551] 

6°54 
11-79 
11°24 

5°08 

8:29 

3°29 


ay. 
litre? atmos. 
unit of mass 
= gram 
mol. 


18-20 
24°58 
30°85 
36°58. 
17°44 
19°20 
26°94 
18°36 
19°95 
25°54. 
11°38 
20°47 
20:24. 

9°53 
17:60 

577 


GU 

litre? atmos. 

unit of mass 
= 1] gram. 


0-C002164 
0:002160 
0.002073 


_ 0:001865 
_ 0001700 
— 0:001923 
- 00005114 


0:0002368 
0:001827 
0'V01312 


| 0:001227 


0:001817 
0001519 
0 001447 
0:004963 
0:002302 
0:01050 


0 
v, in ee. . 


0:09319} 10°73 


06868 1456 
0°6837 1462 
0°6677 1-497 
0°6345 1576 
1:058 09432 
02912 | 3-444 
0'2043 | 4895 
05154 1:940 
04381 2:282 
0:4508 | 2°218 
03727 | 2°683 
0°4590 | 2°179 
0°4523 | 2:211 
04792 | 2:086 
03909 | 2°558 
05048 L981 


on 


gram/ce. | gram/ce. 


12°74 

0°6115 
06142 
06139 
06120 
0°695 

1°4802 
1 988 

0811 

0°9483 
0°9836 
06569 
0-831] 
0°8422 
0°7475 
0:9372 
0°9584 


2°47 
2°39 
2°42 
2°26 


4 08(?) 


2°62 
2°62 
2°79 
2°73 
2°07 


i 
Wi 
‘ 
i‘. 
- 
i! 
To 


| 


* 
i 
te 
ae 
4 


> Se SS SS 


506 Dr. W. C. McC. Lewis on the Nature of the 


It is evident from the foregoing table that the average 
surface densities are distinctly greater than the bulk 
densities—in fact, approximately 2°5 times the bulk density 
for the above-mentioned liquids. 

In view of the apparent generality of the phenomenon of 
increased surface-density (over bulk) one is naturally led 
to inquire how far this may be connected with, or exert an 
influence upon, the molecular surface energy, which in the 
hands of Eétvés* from the theoretical standpoint, and later 
in the hands of Ramsay and Shields t from the experimental, 
constitutes the basis of all our numerical conceptions of 
molecular complexity, or state of aggregation in pure liquids. 
EKétvés showed that the temperature coefficient of the 
molecular surface energy would follow the same curve for 
all liquids. Having carried out experiments with ether he 


found that a ae 


perature, and therefore of course this linear relationship 
should hold for all liquids. Since the expression includes 
the surface tension y it is evident that in the first instance 
at any rate any deductions as to molecular state must be 
applied to the surface layer, and it is rather curious that the 
mean numerical value for ‘‘ normal” liquids comes out 2:1, 


was constant over a wide range of tem- 


ie a 
a number not differing much from the — ratio 2°5. 
mv 


It might therefore be thought at first sight that ‘‘ molecular 
complexity ” was simply a different degree of surface density, 
but that this is not so is amply demonstrated by the fact that 
such characteristically “ non-associated ” (normal) liquids as 
the esters and hydrocarbons (aliphatic and aromatic) have 


practically the same “s yatio as bodies such as water, methyl 


Om 
alcohol, and acetic acid, which are characterized by marked 
association. Further, Hétvés’ original considerations, 


depending as they do upon general principles of corre- 
sponding states, would be practically independent of surface 
density changes as long as the surface was in equilibrium 
with the bulk, which of course is the case f. 


* Kotvos, Wied. Ann. xxvii. p. 448 (1885). , 

+ Ramsay and Shields, Zeitsch. fiir phys. Chemie, xii. p. 433 (1898). 

} The form of Eoétvoés’ expression does require a little modification, 
but the conclusions as to complexity will probably not be altered. 


Transition Layer between Two Adjacent Phases. 507 


The Surface-density of Water and its variation with 
Ti entperature. ' 

This has been measired, as ‘ini the p previous. case, by means 
of equation (3). Fhe temperature range is from us to 100°C. 
The value of “a” is due to van der, Waala*, namely, 3°467 
(units : gram ee atmosphere, litre). When.we take 
the gram as unit this nuxpher becomes 0:0107... fLis- regarded 
as constant throughecat. the range.of temperature, which is 
approximately correct as’ jong -as we -are sufficiently far 
distant from the critical temperature. (The value for a 
found for 100° C. is 0°0105 on the same units, cf. foregoing 


TaBLe I].—Water: Surface density. 


| Eaed RT 

| eee eee ape) 2 
“a Vin= On (cals ) 

361 10000 1-0000 588°4 30:78 
1395 |  0:9993 1:0007 584-74 31:89 
25:14 | 09971 1-0029 580:13 33-11 
37°31 09935 1-0066 574-35 34-45 
5064 «09880 10120 567-26 35-99 
65:36 09807 | 1:0196 558-62 37°56 
81-71 |  0:9707 1-0302 548-19 39°45 
100-0 09587 10431 535°78 41-45 
r: | ] 

Be A | Nip teal Uae oe eae ee ae = 
: * |atmospheres.| 17 ©:¢ | gram/c.c. om 
| | 

361 557-6 2342 | 04568 | 2189 | 2189 
1395 | 5529 | 2321 | 04610 | 2169 | 2171 
25°14 547-0 22°97 | 04658 | 2147 2153 
37°31 589-9 2268 | 04719 | 2119 2133 
50-64 5313 2231 | 04796 2-084 2-109 
65:36 521°1 21:89 04888 | 2-046 2-086 
81-71 508-7 21:36 | 05009 | 1-996 2-056 
100-0 494-3 2076 | 05153 ‘1-940 2-023 


* This value of “a” gives K,,: 10700 atmospheres at 0° C. 
{ Continuitat,” etc.) 


ro 


— T= 


Se FF ES 


508 Dr. W. C. McC. Lewis on the Nature of the 


Table.) Thevaluesof the latent he are those of Winkelmann, 
Wied. Ann. ix. px 833 (1880). 

It will be:seen that o,*and° $i: eenverge slightly with 
rising temperature. This convergence ro doubt becomes 
abrupt as we approach: the: eritical temperature, where of 


—_ o c « Ceo « oto on 
course o,, =a 2 vapor! i 


The A tate ‘Procvure: of «: Water ‘ri he ee of the liquid 
and in thé surface: laver,? and its variation with 


Temperature. 


Having obtained values for the specific volumes in the 
surface-layer and bulk, we calculate the corresponding 
internal pressures by equation (1) or (2). The results are 
given in the following Table. I have here calculated the 
values of K, and K,, first assuming that van der Waals’ 
constant a has the value 0:0107 (and remains constant) ; 
secondly, a is assumed to be 0:0178, its value for the critical 
temperature. The values of K, and K, are no doubt far 
from the truth when calculated on the second assumption, 
since the temperature range 0-100° C. is fairly far removed 
from the critical region (364° C.). Comparison is, however, 
instructive as showing how the want of constancy in a is 
magnified when applied to internal pressure values. If a 
were really constant, K, and K,, should vary only as the 
respective specific volumes. 


TABLE IJ].—Internal Pressure of Water at different 


Temperatures. 
Temperature, K,,= 7 Ks= = K,,= a K,= aes 
)° ©. "m s m Vs 
(a=0:0107). | (a=0-0107). | (@=0:0178). | (@=0°0178). 
atmospheres. | atmospheres. | atmospheres, | atmospheres. 
2°61 10,700 51,260 17,810 30,800 
15°95 10,650 50,350 17,720 30,260 
25°14 10,640 49,320 17,700 29,630 
37°31 10,560 | 48,060 17,560 28,880 
50°64 10,450 46,510 17,390 — 27,960 
65°36 10,290 44,790 17,120 26,920 
81-71 10,080 42,600 16,770 25,630 
100-0 9,840. 41,290 16,370 24,210 | 


Transition Layer between Two Adjacent Phases. 509 


Calculation of the Heat-effect per unit surface area due to the 
surface change in density. 


It is evident from Table IJ. that the surface-density 
approximates more nearly to the bulk density the higher the 
temperature. It follows therefore according to the principle 
of Le Chatelier that the density increase is accompanied by 
an evolution of heat. 

The internal latent heat of vaporization is the amount 
required to remove one gram of the liquid from the surface- 
layer, and taking into account Stefan’s law, the same 
amount of work accompanies the passage of one gram 
from the bulk of the liquid into the surface when new 
surface is formed thereby. In water at 0° C. this heat- 
effect is about 570 cals. Water suffers a density increase of 
1:189 gram/c.c. as we pass from bulk to surface. If we 
take the thickness of the transition layer to be 5x10~° em. 
(Quincke’s value for the average range of molecular action), 
then each unit of surface area corresponds to a surface slab 
of liquid 5x 10-® c.c. possessing a mass of 2°189x 5x 107° 
gram. If there had been no surface-density changes the 
same slab would have had a mass of 5x10-* gram. Hach 
additional sq. cm. of surface is therefore associated with the 
transfer of 1:189x5x10~° gram of liquid. The heat-effect 


‘due to this quantity transferred would be 0:0034 cal. In this 


connexion one might draw attention to the value obtained 
experimentally by J. G. Parks (Phil. Mag. iv. p. 240, 
1902) for the heat given out on moistening powders 
(such as silica) with water. The cause of the heat-effect 
is ascribed to surface-density changes in the water where 
solid and liquid meet. The heat evolution per sq. cm. of 
powder is 0°00105 cal—a number which is of quite the 
same order of magnitude as that calculated for the case 
water/vapour. 


The general structure of the Surface-layer. 


Willard Gibbs in his celebrated memoir was one of the 
first to point out that a surface-layer is in general hetero- 
geneous, as, for example, oil in contact with water. The 
density changes of each phase do not consist in a steady and 
rapid fal) or rise as we traverse the layer from one side to 
the other, but if the considerations put forward in this paper 
are correct the density of each phase will pass through a 


510 = Transition Layer between Two Adjacent Phases. 


maximum before falling to zero. The water-oil case might 
be represented thus (fig. 1) : 


Fig. 1. 


I 
t 
{ 
I 
t 
4 
{ 
( 
t 
{ 
( 
' 
’ 


The vertical dotted line represents the imaginary mathe- 
matical “‘ surface” ; the distance AB represents the thickness 
of the heterogeneous layer, the horizontal lines representing 
the constant bulk densities on each side. For the case of a 
liquid in contact with its own vapour, we obtain (fig. 2): 


Fig. 2. 


Liquid WATER / 


‘ 
' 
1 
1 
! 
‘ 
| 
| 
) 
i 
! 
1 
| 
' 


in 
* 
| 


’ 
' 
i 
1. 
! 
' 
' 
' 
1 


' 
i] 
t 
1 
t 
f 
! 
} 
| 
i 
i 
i} 
i 
1 
| 
1 
! 
1 
i] 
: WaTER Varour 
i] 

1 

1 


] 


Ax——>'B 


1 ' 
' i} 


The magnitudes denoted by o, and K, are the average 
values over the layer as a whole—the actual values over any 
small portion of layer may be considerably greater or less 


than these. 


Summary. 


1. The values for the internal pressure in a liquid, as 
obtained by the Bakker-Dupré method of calculation, are 
considerably larger than those calculated from van der 
Waals’s equation. A cause for this is sought in the assump- 
tion that the density in the surface-layer is greater than in 


the bulk. 


ee ee ee ee ee 


| 
; 


On the Lagging of Pipes and Wires. 51k 


2. A table is given showing the average values for the 
surface density of a number of liquids at their boiling- 
points. 

3. A table is also given showing the variation of the 
surface density of water with temperature ; the temperature 
variations of the internal pressure, both in the bulk of the 
liquid and in the surface-layer, are likewise tabulated. 

4. The heat effect involved in this density change is 
calculated per unit area of surface and is found to be 0:0034 
cal.—a quantity which agrees approximately with Parks’ 
analogous determinations. 


I should like to take this opportunity of expressing my 
indebtedness to Prof. W. B. Morton, the Queen’s University, 
Belfast, and to Prof. A.W. Porter, University College, London. 


LEON the Lagging of Pipes and Wires. By AuFrep W. 
_.Portsr, B.Sc., Fellow of, and Assistant-Professor of 


+ Physics in, University College, London ; with an Addendum 
| in conjunction with Mr. E. R. Martin, B.Sc.* 


| ae effect of surrounding a pipe or wire with a lagging 


material does not seem to be perfectly understood. A. 
coat of poor thermal conductivity may keep a hot pipe cooler: 


than it would be without such a coat. The problem occurred 
to me a few years ago in working out examples to set toa 
class of senior students ; but it is also a problem of practical 
importance, especially in connexion with laboratory apparatus. 
On looking the question up I find only a reference to such 


an effect in a paper by Professor Bottomley (Roy. Soc. Proc. 


vol. xxxvii. 1885), where it is shown experimentally that a 


difference of a degree or so can exist between two equal wires 


carrying the same current, one of them coated with a thin 
coating of various materials such as shellac, while the other 
is bare ; the coated one being the cooler of the two. Such 
experiments as these give, however, an entirely inadequate 
notion of the possible magnitude of the effect as the sequel 
will showt. A striking experiment is to take a thin platinum 
wire and coat it at intervals with a thin layer of glass— 


which is easily fused round the wire. If a suitable current. 


be now passed through the wire the uncoated portions may 
* Communicated by the Author. 


+ From a remark in Professor Bottomley’s paper I gather that. 
Lord Kelvin was aware of the anomalous results to be expected. The- 
matter, however, seems to have dropped out of sight owing to the lack. 


of success in obtaining any experimental support. 


512 Prof. A. W. Porter on the 


be made nearly white hot, while those parts which are covered 
with glass do not even glow. Insuch a case there are several 
hundreds of degrees difference in the temperature of neigh- 
bouring portions of the wire. This experimental result 
accounts for the platinum wire sealed into an ordinary incan- 
descent Jamp remaining as cool as it does. 


The problem in its sinyplest form. 


A cylindrical wire of good conducting material and radius 
a is surrounded by a concentric sheath of radius 6 and thermal 
conductivity k The wire is heated by means of an electric 
current or otherwise to a constant uniform temperature by a 
rate of supply of heat H per unit length of the wire. Then 
if the thermal emissivity from the outside of the sheath is e 


we have 
es SG, OE ashe Hit 00 
H=-—k.2a 5.) =k dans 


=e. 2rb6,, 
where @ is the temperature of the outside of the sheath, 
OD a8 
and (x) is the slope of temperature in the sheath at its 


inner surface. The corresponding solution for the tempera- 
ture of the wire is 


where 9, is the temperature of the wire. 
To find the effect of changing the thickness of the coating 
we differentiate this equation with respect to b: giving 


O@~,/7 EL 1 aS 
Ob ~ Qa | Het BB 
The effect then is to increase @2 (for a given rate of supply 
of heat) provided that 6 is greater than A/e; the coating 
under these conditions acts asa lagger. But if 6 is less than 
this critical value the effect of its increase is reversed ; the 
coating under these circumstances promotes the outward 
flow of heat. This critical value if it exists is independent 
of the radius of the wire ; but it must be noted that since b 
can never be less than a, it is possible that for a particular 
wire it does not exist. To find out the likelihood of its 
occurrence we must examine the values of & and e which 
~will probably enter into account. 


Lagging of Pipes and Wires. D13 


The old values of the thermal conductivities of badly 
conducting materials still given in most text-books are mainly 
due to Forbes. They have been shown, however, to be com- 
pletely untrustworthy. The values employed here are those 
obtained by Dr. C. H. Lees by Sir O. Lodge’s method 
(Trans. Roy. Soc. A. vol. 183. 1892, p. 481). 

With regard to the emissivity there is more indefiniteness 
as it depends, when the body is in air, not only upon the 
nature of the surface but upon its radius, as was first shown 
by Peéclet. Provided, however, that we do not consider 
sheaths whose radius is very small, we may take ‘0003 as a 
fair value. 


The critical values of 4 are then as given in the following 
table :— 


Tas_eE I. 
Mite Thermal Critical radius 
conductivity. | in centimetres. 
ere AIS ...........-0es sean eee ene | *00243 | 871 
on 5 0047 | 16 
OS I Sa nee es | 0006 2 
“ors oie Sy 00088 | 1:3 | 
SRECREGR-ELCNA ....5..2....0-.ecenu seas aaetnes| 00046 1-5 
ERB ten nic a. 0002+ asgn5<omae eee 00031 1:0 | 
eesrOs PAPEL ..........-.ce<c<snnqeceeeemeees 00057 19 | 
TIES (2. ase So ) "00013 “4 
ip | a | 00022 | oe & 
ii oa eeecemepee es ke 00055 18 
Lin 10: NR 82 | 00023 8 | 
*Magnesia (Pattinson’s Light Calcined)... ‘00016 9) | 


* Hutton & Beard, Faraday Society, July 1905. 


Here we have the somewhat startling result that coating a 
wire with glass up to 8 cms. radius is more and more detri- | 
mental to the maintenance of a high temperature in the wire 
the thicker the coat is within this range. For gutta-percha 
the range is up to 1$ ems., and so on for other substances. 

If the coating has an outer radius greater than the critical 
-one the action begins to reverse. But an examination of the 
formula shows that the improvement (from the point of view 
of effectiveness in lagging) is very slow, depending ultimately 
upon a logarithmic term. The result is that very considerable 
thicknesses must be attained before the temperature of the 
wire will become the same as if there were no lagging in it 
atall. This point is reached when 

1 1) ia tape: 1 
be he eee 


Phil. Mag. 8. 6. Vol. 20. No. 117. Sept. 1910. 2M 


SS - = SSS SS SSS Se 


514 Prof. A. W. Porter on the 


This second radius, which may be called the neutral radius 
for the particular bare wire, because for it the total tempe- 
rature effect of the lagging material is zero, is not independent 
of the radius of the wire. It is not possible to express it, 
therefore, quite so simply. It is obvious, however, that it 
corresponds to the value of 6 for which the expression 
s ial 
be * k 
then the value of this expression be calculated for various values 
of & and be plotted against b, then the radius of the bare 
wire and the corresponding neutral radius are the two values. 
of 6 for which the ordinates are equal. Corresponding values. 
so read off from a curve are tabulated with sufficient accuracy 


in Table II. 


log 6 has the same value as for the uncoated wire. If 


TABLE [].—Para Rubber. 


k=°0004. e= (0004. 
| 
Radius of bare wire. Neutral radius. | 
*25 em. 10 cm. 
on 5'8 
"50 | 2°9 
1:00 Critical 1-00 


In order to test these results a thin platinum wire (02 em. 
radius) was coated along a part of its length with glass so 
that the outside radius was‘l1cm. When a current is passed 
through the wire the uncoated portion may be made nearly 
white-hot without any sensible glow occurring in the coated 
portions. This is the nature of the result to be expected 
even from the above simplified theory, for it gives 


Temperature excess of uncoated wire _ 1 
Temperature excess of coated wire ~ /1 1, 6)’ 
(5 +7, 1085 
c 


where 6, =the critical radius. | 

In the above case this ratio is 5 nearly. It should be 
observed in passing that the logarithmic term scarcely affects. 
the ratio, and this will usually be so when @ is small compared 
with the critical radius. 

Thus if the uncoated is raised to 1600° C. the coated will 
be only at 330° C. 


Lagging of Pipes and Wires. 515 


Deviations from simple theory. 


Where the difference of temperature is so considerable it 
is of course not sufficient to consider the various data as 
independent of the temperature. In this experiment the 
same current passes through both portions of the wire, and 
therefore, owing to the change of specific resistance with 
temperature, the rate of generation of heat in equal portions 
is not the same. If & is the coefficient of increase of re- 
sistance with temperature, py the specific resistance at 
atmospheric temperature, and C the current, the equation 


becomes 
_ po 1 +262) C? e! be iseb 
ba = Oat bet BG f 


27a? 
or 6 


Lea 


The critical radius is still b=4/e, but 6,/(1+6,), say 9, 
takes the place of @,. Hence for the given wire and coat 
we have: 


ees 
AY + qlogz b. 


a - 


@ancoated] Outed —9 nearly 3 


and if a be taken as ‘004 this gives a temperature of about 
50° C. for the coated portion when the uncoated is at 1600° U. 
The increase in the resistance with temperature has thus a 
very large intensifying action. The value of (1+°004 6.) 
may be called the “intensifying ratio” due to the resistance 
change. Its value for different values of @, is given in 


Table III. 


TABLE ITT. 
0a | 1+ 004 0a 
0 1 
100 1°4 
500 3-0 
1000 5:0 
2000 9-0 


That is to say,a current which would maintain a wire at 
200° excess with a given coat if the resistance remained 
constant (as it would do approximately in the case of an 
alloy) will heat it to 1000° C.in the case of a metal for 
which «='004 (as it ar is for pure metals). 


516 Prof. A. W. Porter on the 


i Eaperimental, 


i It was considered to be of value to determine the actual 
if temperatures acquired by the different portions of the wire. 
i For this purpose a wire was prepared with potentiometer 
leads A, B, C, D, arranged as in fig. 1. 


i Fig. 1. 

| Va Vz Glass Vo Glass Vp Glass 

il ‘ 
nmr : d 

i i | | 

i eae 

i The two short lengths of glass-covering eliminate the end- 
Mi effects of the coated portion. The distance from A to B is 


made as nearly as possible equal to that from C to D. The 
differences of potential V,—Vz and Vce—V,y when various 
steady currents (measured by a shunted Sullivan galvano- 
meter) flow in the wire were determined by means of high 
resistance galvanometers which were calibrated for the pur- 
pose. The temperature coefficient of the platinum wire was 
determined, also the value of V,— Vs corresponding to a 
measured small current—the wire being immersed in oil— 
in which case the wire was assumed to be at the temperature 
of the oil. From these data the temperatures of the two 
portions of the wire were calculated. There is no need to 
give the readings as only rough values were aimed at. 
The temperature-excesses (above 18° C. which was the room 
temperature) are given in Table IV. 


TABLE IV. 

Uncovered. Covered. Ratio. 
832 152 5:5 
702 | 117 6:0 
1]2 32 3'5 

87 15 t 58 


| The radius of the bare wire was ‘015 cm. and of its covering 
i *235 cm. 

i Great as the ratio of excess temperatures is, it is not nearly 
| t ber as the “simple theory ” requi iz. 15 
| as great a number as the “simple theory” requires, viz. 15. 
i Still less does it come up to the requirements of the theory 
1 when allowance is made for resistance change. Similar 


Lagging of Pipes and Wires. , SLUT 


observations were made on a nickel wire partly coated with 
gutta-percha. ‘The results in this case are given in ‘I'able V. 


Taste V.—Nickel wire and Gutta-percha. 
a='025 cm. b='175 cm. 
Hxcess temperatures. 


Uncovered. Covered. Ratio. 
ats 

19 14 . 1:36 
31 18 | 1-70 
3 25 1°44 
65 | 27 2:4 | 
67 24 27 
77 | 42 18 

111 : 51 2:2 

| 


| 


The simple theory would require a ratio of about 6. 

The cause of the discrepancy is to be sought for in the 
variation of emissivity and thermal conductivity with the 
temperature, and in the case of emissivity with the radius 
also. The last cause is in fact exceedingly important in the 
case of wires so fine as those used in these experiments. The 
ratio @a/©, should in fact be more nearly : 


b 
mae ie 
than 4 where EH, is the emissivity for the covering and H, 


is that for the wire. In the case of the platinum wire at 
~ about 100° excess E, is perhaps as much as 7 times E,, hence 
a ratio of 15 is at once lowered to about 2. 

The full and satisfactory consideration of the question 
requires, therefore, a complete knowledge of the variation of 
the various data with temperature. In the absence of this it 
scarcely seems worth while to develop the theory much 
further owing to the increased complication. The following 
_ formulation may, however, be found useful in case the 
question should be found of sufficient practical importance. 

The differential equation to be solved is 


,09 ___ H 


Of on ene 


a 


———- 


a 


SSS 


Sa CSE Ce ee Ses 


Sas 


018 Prof. A. W. Porter and Mr. BE. R. Martin on the 


the solution of which is 


K,—K,= Fe og ~ 
where ie Ne kdé. 
Differentiating with regard to 6 we have 
dé nO, eee 
hae Fs ap haga 
Hence the critical radius is given by 
Pi Mt) gut 
db 2arbky ’ 
or since H=e .2rb0o, 
de ae (5) 
sD ob Key - 
where d 0. dé; 6 


db — Ob si db 00,’ 
the values being those for 6 equal to the critical radius, and 
therefore 
QB, i, cel 
db) Barbky” 


If we can neglect the variation of e with the radius at 
constant temperature as we can do when the radius ceases to 
be very smal], the critical radius becomes 


oe 2 ( jee oe. 
nae €b 2erezky, an 

Thus the critical radius is seen to depend upon the rate of 
heat supply. The value in the simple theory is the value for 
a very slow supply of heat. 


The above is worked out on the assumption that the value 
of H is constant. 


ADDENDUM (in conjunction with Mr. EK. R. Marrin). 
The lagging of steam-pipes. 

A similar theorem to that proved above is, of course, valid 
also for the case of the condensation of steam in pipes. The 
amount of condensation is proportional to the escape of heat 
from the surface. We have as before 


Hy b 
es slay +- ; be). 


Lagging of Steam-Pipes. 519 


In order to test this equation we have taken two thin 
brass tubes, each 30 cms. long and *19 cm. external radius. 
These are inserted in a steam-trap (fig. 2) so that steam 


- ~~ 
(f To 
gun Boirer 


issues simultaneously from both. One of the tubes is coated 
in succession with increasing thicknesses of asbestos paper, 
while the other has throughout a single coating of asbestos 
paper, the object of this being to make the character of the 
surface as nearly as possible the same for both, and thus 
promote an equality in the values of the emissivity for them. 
Under the orifice of each tube is placed a collecting test- 
tube into which drops the steam which is condensed in the 
pipes. The amount collected from the more thickly coated 
- pipe varies with the thickness of the coating. By taking the: 
ratio of this amount to that collected from the thinly coated 
one, a value is obtained which may reasonably be taken as 
being more independent of varied conditions of the flow of 
the steam than if the value for a thin coating were deter- 
mined once for all. The two tubes when coated each of them 
with a single layer did not correspond to equal condensation 
owing to some small difference of circumstance. The ratios 
obtained have subsequently been raised 5 per cent., so as to 


520 ~=Prof. A. W. Porter and Mr. E. R. Martin on the 


change to unity the value corresponding to a single coat.. 
This is merely equivalent to changing the standard to one 
which is more convenient for calculation. Several observa-- 
tions were made for each thickness ; in the following table 
the mean values are alone tabulated :— 


| Ratio of condensations | R 


| No. of Coats. pee Thick coat. Ditto 
; Thin coat. Reduced standard.| 
1 ‘234 ems. ‘953 1:000 
2 Ss, "984 1:032 
4 318 1-068 | Ti22 
4 °350 1:084 | 1:140 
5 397 1:099 1°154 
6 447 1°155 1:214 
| 8 ‘514 1:234 | 1:296 
| 10 ‘580 et 1°272 
| 12 ‘671 1:180 1:239 
14 “749 1185 1:244 
18 “890 | 1161 bee Al bg 
sa 1-194 | 1:070 1:125 


The theoretic value for the ratio should be (assuming that 
the temperature at the external surface of the brass may be. 
taken as constant throughout the experiments) 


ge 
‘9346 * & 928" a9 
fy ee ae 
bE TR 2" 79 


Ratio= R= 


where "19 is the radius of the tube uncoated, 
"234 “p i with one coat, 
EK and & are emissivity and thermal conductivity of 
asbestos. 
b=external radius of coated tube. 


Now E and & both depend upon the excess temperature, 
and EK depends also upon the radius. The experimental 
results are not accurate enough, however, to justify one in 
trying to take these variations into account. Assuming: 
constancy in these data, it is easy to show that R should 


be a maximum when 0= * Now from the diagram the 


maximum is found to occur when 6=°57 cm. (about). The 
position of this maximum fixes the ratio of k to E. Putting 
the value of k derived from this value of 6 into the equation, 


Lagying of Steam-Pipes. 52k 


the result is independent in any further way of both & 
and E ; the equation in fact elie 

1 2a ae 
934 F 57 °8°-19" 

1 1 - 
5 + 57 loa ag 
Thus it is unnecessary to know the absolute values of & and 
H, though we shall discuss these later on. 

This theoretic equation is represented by the continuous 

curve on fig. 3. It will be seen that it corresponds with a 
fair amount of accuracy with the experimental values. 


Fig. 3. 
pele 
ts O 


R= 


Theoretic Curve 
Experimental Poi 


id it 
ne oe a 


It was considered worth sacl if obtain the absolute saliee 
of the emissivity and conductivity of the asbestos paper 
employed. The former was obtained from the rate of cool- 
ing of a thinly coated bar, the temperatures being read by 
the thermoelectric potentiometer method. Plotting a curve 
of temperatures against time, and finding oraphically the 
slope at the 100° point of the curve, the emissivity was cal- 
culated from the slope and the dimensions of the bar. The 
value so obtained was ‘000275. 


5922) Mr. D. Tyrer on Relations between the Physical 


The thermal conductivity was measured by the slab method 
—a slab of about 300 sq. cm. area and of a thickness of 
*245 cm. being pressed between a steam and a cold-water 
vessel. The temperatures of the cold water were read from 
minute to minute as they rose. Careful corrections were 
made for “ radiation ”’ loss from the cold vessel. Two distinct 
sets of readings were taken. The values of the conductivity 
obtained were ‘000150 and :000141 respectively. Taking 
the mean of these and employing the value obtained experi- 
mentally for the emissivity, viz. "000275, the critical radius 
becomes °527 cm. The value read off from fig. 3 is about 
-o7. The agreement is as good as was to be expected. 

The conclusions to be drawn from these experiments are 
that : 

(1) On narrow steam-pipes, up to about half acm. ex- 
ternal radius, the application of an asbestos coating encourages 
the escape of heat. This radius is below what may be termed 
engineering dimensions. But in experimental apparatus it 
is not at all an unusually small radius ; and for such pipes 
the application of lagging is a delicate question. 

(2) When the radius is much smaller than this critical 
value, the coat must have an external radius much greater 
than half a centimetre before the lagging efficiency is as 
good as without a coat at all. Thus, the curve would seem 
to show that the escape of heat is the same for external radii 
of -24 cm. and 2:0 cms. 


. LIL. Relations between the Physical Properties of Liquids at 
the Boiling-Poit. By D. Tyrer *. 
SIMPLE relation exists between the latent heat of 
| vaporization of a liquid and the molecular volume at 
the boiling-point. The relation may be expressed 
iM ke 
where L is the latent heat of a liquid at its boiling-point, 
V the molecular volume at the boiling-point, M the molecular 
weight, and K a constant, This relation, which may for the 
present be regarded as empirical, holds with a fair degree of 
accuracy for most classes of substances. In the Table given 
below the values of the molecular volumes are the mean 
values of the results chiefly of Kopp, Pierre, Schiff, Thorpe, 
Gartenmeister, and Young. In the last column are given 
the latent heats calculated according to the above equation, 
the value of K having been taken as 1583. ‘This figure 
is the mean value of K for the aliphatic esters, calculated 
from the results of Schiff and J. C. Brown. As Brown’s 
* Communicated by Prof. H. B. Dixon, F.R.S. 


Properties of Liquids at the Boiling-Point. 


523 


results are invariably higher than those of Schiff, the 
agreement between the culeulated and observed ae of 
ihe latent heat will not be so good in those cases where the 
results of only one of the apouel observers are known. 


Molecular “ - 
Volume V| Latent atent 
Liquid. at the Heat Authority. | Mean. Heat 
Boiling- | (observed). (calculated), 
Point. 
Propy) formate.............i.0s. 107°0 {9036 na 87°89 85°41 
Isobutyl formate ............... 129°9 { ae eae 78:56 | 78°52 
Isoamyl formate ............... 151 | | a cae (2°70| 72°60 
98:26 Brown 
Methyl GLEE Y 1: a A ne Son | 97-0 Ramsay & 97°63 93°57 
es Marshall 
Ethyl acetate 2... | 1060 | {Sar | Brown | 8273] 8518 
| vs 
Propyl acetate .........-.....e- ; Pee | 887). raat 
EME ACELALG........5.0c.0c0800c4] 1506), ) 739 Brown 73°9 72°65 
Isobutyl acetate ............... b 1507) {70:46 Sa CLS) T2aG 
nas i | 
Isoamyl acetate.................. | L4 | { 69.00 ail 67°67 67-94 
Methyl propionate ............ | TOLG is oaa ae 86°57 | 84°72 
Propyl propionate ............ 1500 | { Ey ola 72°61 72-46 
Ethyl propionate ............... 127°8 { a a 187 78:72 
Isobutyl propionate ............ arcing 66:0 Schiff 660° |. 674 
Isoamy] propionate ............ 196 : | ene Se aie 6418) 63:8 
Methyl butyrate .......0..0... wer | {7h7? Bee 3) 7800|. (2807 
‘Ethyl butyrate... | 1504 | {4365 | Brown | 7257). 7282 
Pro 1 b 3:9 66°2 Schiff 9) S 
py butyrate) ..2.....6-./.- 7 | 4 68-29 eee 67:24, 67°98 
Propyl isovalerate ............ 192:2 64:37 Brown 6437, 63:91 
Tsoamyl butyrate ........-..+-- OPT Ml ey aati \ GEES), WoO ebay antes 
Bett ul pityratont ce 199-2 { bere Sega Las. ace 
Methyl isobutyrate ............ W205 ES fiemem DH einew elle thE nn TOM 
: 
| Ethyl isobutyrate.........0... OT Ete een, (TORE tee 
| Propyl isobutyrate ............ 174 | 63:9 Schiff 63:9 67°94 
Tsobutyl isobutyrate........... 1902 ed gut eda, | O82 [ox OMNG 
| Isoamy] isobutyrate ............ | 223-0 57°65 Schiff 5765 60°72 
Methyl! valerate.................. i 149-1 69-95 Schiff 69:95 | 72°31 
Isoamyl valerate ............-. | 55 B62 Schiff | 562 | 57-2 
| 


024 Mr. D. Tyrer on Relations between the Physical 
: ETT 
Molecular 
Volume V Latent 
Liquid. at the Latent Authority, | Mean. Heat 
Boiling- Heat | (calculated) | 
Point. | (observed), 
Methyl isovalerate ............ 148°7 | 72°38 Brown 72°38 7225 
Ethyl isovalerate ............... 173°3 | 67°84 Brown 67°84) 67°85 
Ethyl caprylate ........0.0.... 1977 | 60-46 Brown | 60:46] 63:99 
Isobutyl isovalerate ............ 222 60°41 Brown 60°41 60:63 
Haliyl nonylate 2.0... ..........-5 ee non 58:08 Brown 58'08| 56:0 
Diethyl carbonate............... 138°8 72°85 | Louginine | 72°85} 68:85. 
ROOF ioeie ce cietioaicigs nsnini’s » paiva 1089 54°1 Bertheiot | 54:1 51:10 
IOP GORGE (C00), oi. n0seses acs 84:5 58°49 Wirtz 58°49| 58:09 
Carbon tetrachloride ......... 103°7 46°35 Wirtz 46°35 52°34 
tas , 46°87 Andrews Ae te 
Biphryl iedide ....../.s.escc060s.2 86"1 | 460. | Kahlenberg | 4643] 4454 
Methylene chloride ............ e512 | 753 |{ ee & 75-3 | 74-87 
A ae ( 60°37 Wirtz d , 
Ethyl PROUUIOS EP. hcccse sss ase ‘i 7 | 61-65 Berthelot 61:01 61 98 ! 
Methyl iodide ...,....-...is.-« 63°9 46°1 Andrews | 46:1 44°54 
PUGEUMIMEM YOO). 05.066. ssdennee 56°9 136°4 Berthelot |136°4 138°3 
fe Griffiths & 
Benzene 95-9 oe, | Marshall | o4g7| gos 
CO i oO 94:93 Brown a 
92°91 Wirtz 
Hithyl benzene <2 .........006000. 138°8 76°4 Schiff 76°4 TOST 
, { 66°3 Schiff oie oF 
ROMINA 2 sc ew sl. deckenssaeees 184-4 | 67-64 Bean 67-0 67:20 
: nc} 71°75 Schiff : 
AVGRIEVICTIO... focot esses ces esevees 162°4 { 74:42 Brand 73°08; 71:92 
EerOpy! DENZENE ......... 02.02 150s 161°8 71°75 Schiff T175| 71°83 
68-73 Favre & 
PRC EMCUEN G52 2h ee giie «shies sakes 182°8 Silbermann | 68°61 66°01 
68°5 Schall 
83°55 Schiff 
<i : Ramsay & ; : 
MIGIMENOS «2... 1.66 A dee ceer cose: 117°9 86°8 Atcha 85°92] 84:34 
87°43 | Brown 
Dak iE ee 1388°7 82°47 Brown 82°47 17°25 
Beenie) dia cee, fo. anat ” 81:34 Brown : ; 
BP RVACHE 2 8G iacdesasvest as 139°7 { 78-25 Schiff T977| = 77-44 
BEAMS 29285. i2o nae owapizee 140-2 80°98 Brown 80°98} 77°53 
Piperidine......... yoeeeees tele 106°3 88:9 Louginine | 88°9 88°8 
Sulphur oxychloride (SO,Cl,) 86°3 52°4 Ogier 52-4 51°79 
Pin tetrachloride ............... 31-2 30°33 Andrews | 30°33] 30°91 
Silicon tetrachloride ......... 123°4 373 Ogier 37°3 46:3 
Carbon disulphide ............ 62°71 833i |) Wirtz 83°81 82°43 
Phosphorus trichloride ...... 93°4 51°4 Andrews | 51-4 52°4 
Ethylene oxide .................. 51-8* 138°6 Berthelot |13886 | 1341 
Ethyl propyl ketone ......... 143°5* 82°96 Louginine | 82°96} 82°78 
Dipropyl ketone ........... ... 165:3* 75°94 Louginine | 75°94| 76:17 
Methyl ethyl ketone............ 99-3* 103-4 Louginine {103-4 | 1017 
Pseudocumene: .......-.2. +25... 176* 728 Schiff 728 72°67 


* ‘These values of V are calculated from the atomic volumes of Kopp, 
experimental values not being obtainable. 


Properties of Liquids at the Boiling-Point. 525 
ls Molecular | 
Volume V__Latent | Latent 
Liquid. atthe , Heat Authority. | Mean. Heat 
Boiling- | (observed). (calculated). 
| | Point. | | 
a Heca estan | wid 
| , ; Mabery & } 3 
| Miemeptane 92.012... 16254 740 ; Gosia 740 | 86-33 
, =O: abery & eQ. see 
| BERANE, os oa: nine de «'dydin sphoiaeip te 139°8 79°4 { Giildstein 19°4 93°3 
| ( 70°92 Louginine 
ROMPMIAMIG. oo... ecdanccchesapee| 18615 74-4 Mabery & | 71°01 79°23 
Goldstein 
| | 
se . f { Mabery & lox ane 
BREIMOR MY)... .......cc0ccecsecracs] | AOL O | 60°85 | Goldstein 60°85 68°35 
: | f 84-74 Brown j “a 
Pilylether ....:.2.csce0..sase 1061 { Be ON dearer)» 8458| | F0L8 
‘ Favre & b 
ee 246 | 69-4 { Sr One ance 
Inquids whose Molecules Dissociate on Vaporization. 
Berthelot | 
2.6) 
Water 18°78 ies Favre& 536-0 | 233-5 
eee aeseererccstenssssseves 585°97 Silbermann | oD 
| (585°9 Andrews | 
262°2 Brown 
Methyl alcohol ..........6...00-. | 42-6 261:7 Schall | 2625) 172°7 
| | 268°7 Andrews 
| ( 202°4 Andrews | 
Ethyl alcohol .......e..cs0e- 62-2 Se ey (20rd |. 1862 
216°4 Brown 
: 166°3 Brown. (| 4a). s 
meeropyl alcohol ..............; 81:2 162-6 Schlamp 164°4 1141 
120-7 Favre & 
PRP BCU occ es eae * ans sesene 41:1 Silbermann | 120°5 118°7 
120°37 Brown 
. ee” 
PR CEUMNACIO «nic sinictlecoanvesessoees 636 | 89:79 | Louginine 90°6 105-2 
| | | 97°05 Brown 
| | ; Favre & 
(ES SOOT ee 108.13 Tf 68 «| Sitbermann | 114-3 |) ).85-6 
| (11396 | Brown 
BRE CHONG ory geSaucicestesmaanbwesy 17°3 | 125°3 Wirtz 1253 116-2 
PR TOUS 0.8 ance nes con eo 150°4 | 47°5 Berthelot | 47°5 42°5 
Diethyl oxalate...............0. 166°2 12°72 Andrews | 72°72 598 
| ( 116-1 | Andrews & 
Methyl formate ............... 62:7 Ogier 113-2 104°7 
i 110°45 Brown 
| 100°1 Brown 
4 Exty) formate: oo. i.0s20.6.00c0- 84°7 100°4 Berthelot & | 1002 93°9 
| Ogier 


526 Mr. D. Tyrer on Relations between the Physical 


It will be observed from the Table that there is a close 
agreement between the calculated and observed values of 
the latent heat. The only classes of substances which do 
not seem to conform to the relationship are the aliphatic 
hydrocarbons and ethers and of course associated liquids 
as the hydroxyl compounds. ‘The validity of the relation is 
in general affected by the following factors :-— 

(1) Experimental errors in the values of the latent heats. 
These are rather considerable as a comparison of the results 
of different observers shows. The difference between the 
calculated and observed values of the latent heat does, as a 
general rule, come easily within the range of experimental 
errors except in the fews cases mentioned above. 

(2) Association of the molecules and their dissociation on 
vaporization. 


The Liffect of Molecular Assaciation on the Relation. 

The last portion of the above Table deals with liquids 
which show molecular association, and it will be observed 
that in these cases there are very wide divergences between 
the calculated and observed values of the latent heats. The 
question may be divided into two cases, viz. :— 


(a) Liquids whose molecules are associated but which do 
not dissociate on vaporization. 
(b) Liquids whose molecules dissociate on vaporization. 


Case (a).—In the first case the relation should obviously 
be written | 


LMn=K V/V Vn, 


where n is the association factor. 
The true latent heat then becomes 


pee ee 
Mni 

Whereas the latent heat L, calculated according to the 
equation without considering association is 
K VV 

i, 

For this class of liquids, therefore, the calculated values of 
the latent heats will be greater than the observed values. 

Case (b).—The second case includes liquids like water, 


whose molecules are associated in the liquid state, but are 
normal in the vapour state, and also liquids like amy] iodide, 


Ci 


Properties of Liquids at the Boiling-Point. 927 


which on vaporization suffers a partial chemical dissociation 
into amylene and hydriodice acid. 

Suppose that the process of vaporization takes place in 
two stages, viz.:—(1) The vaporization of the associated 
molecules without dissociation ; (2) the dissociation of the 
associated molecules of vapour. Let the heat absorbed per 
associated molecule in the first stage be HMn, where H is 
the specific heat absorbed, M is the molecular weight, and 
n is the association factor. This will be equal to K V Vn. 
For the second stage let the heat absorbed per associated 
molecule be XMn, where X is the specific heat absorbed. 
Therefore in the complete vaporization of one associated 
molecule the total heat absorbed (apart from heat required 
to do external work) may be written 


HMn+xXMn=K / Vn +XMn. 


And if X represents the heat required to do external work 
during the second stage of the vaporization, the true latent. 
heat L, of vaporization becomes 


L,= g7 {K VVn+XMn}—2, 


KVV 
= Maan +X—nx. 
Whereas the calculated latent heat not considering molecular 
association 1s 


{ey 
M 


Therefore we may say that for liquids where molecular 
dissociation occurs on vaporization, the latent heat calculated 


according to the equation LM=1583 VV may be either 
greater or smaller than the observed latent heat according to 
the magnitude of the heat of dissociation. 

It may therefore happen that an associated liquid might. 


not show any deviation from the equation LM=K VV. An 
example is formic acid for which, as it will be noticed from 
the Table, the calculated value of the latent heat is very close 
to the observed result. Water and the alcohols give calculated 
values lower than the observed, whilst some of the acids 
give higher and some lower calculated values. 

On the whole it may be said that with the exception of 
the aliphatic hydrocarbons and ethers, the relation holds 
for normal liquids with an exactitude which quite corre- 
sponds to the exactitude of the latent heat determinations. 


928 = Mr. D. Tyrer on Relations between the Physical 


Indeed, by the aid of the equation LM=Kv,/V, the latent 
heat of a liquid (other than those excepted above) can be 
calculated merely from a knowledge of its formula with a 
very fair degree of accuracy. In the above table the values 
of the molecular volume for several liquids marked with an 
asterisk are calculated from Kopp’s atomic volumes, and 
it will be noticed that in these cases there is a very fair 
agreement between the calculated and observed latent heats. 


A Relation between the Molecular Volume and 
the Boiling-Point. 


If the relation LM=K </V be combined with Trouton’s 


equation LM=20°5 T we get a very simple equation 


T=K, VV, 
where T is the temperature of boiling, V is the molecular 
volume at the boiling-point, and Ky is.a constant. 


The value of the constant in Trouton’s equation varies 
for normal liquids from about 19°5 to 22°5. Taking the 


constant in the equation LM=K VV as equal to 1583 we 


should expect that the value of K, will vary roughly between 
70 and 80, and will have a very small value for the aliphatic 
hydrocarbons and ethers. 

The validity of the equation is tested in the following 


Table. The values of T and V are the mean results of Kopp, 


Pierre, Buff, Thorpe, Zander, Gartenmeister, Young, and 


others. 
Temperature Molecular Top 
Liquid. | of Boiling Tgp | Volume V at | K,= -3 =: 
(absolute). Boiling-Point. VV 
Aliphatic Esters, 

thy! formate <2... .cabenk 327°3 84:7 74:53 
Methyl acetate .................. 330°1 83°7 75:47 
mop yl PORMMAEE «60 .cecesnsce sa: 353°9 1070 74:54 
Biehyl acetate yp) iw fel. es, vest 350°1 106°0 73°98 
Methyl propionate ............ 352°7 104°6 74:95 
imubyl) formate i... 372 130°7 73°2 
Isobutyl formate ............... 370°9 aL ISS 73°2 
Propyl acetates... cecwnses. 38745 128°4. 74:2 
Ethyl! propionate ............... 372 1278 73°8 
Methyl butyrate oe 3797 1267 748 
n., Amyl formate ............... 403°4 150°5 758 
Tsoamyl formate .............. 396°7 151°0 74:5 
BDthiylibuhyrate .. ....6b.c0e.s. 04 393 150°4 739 
Methyl valerate ........./..... 399°3 149°] me (595) 
iy Atmiyl acebate’ oe... ca. 420°6 173°8 75°4 


Properties of Liquids at the Boiling- Point. 529 


Temperature Molecular Trp 
Liquid. of Boiling Tsp | Volume V at | K,=—=. 
(absolute). Boiling-Point. A/V 
Isoamyl acetate ...........0668 409 174 73-2 
Ethyl valerate ..............0.68 aie 1745 7438 
Isoamyl propionate ............ 433°7 196-0 T47 
Butyl butyrate 1.9. 3.sdec.-.c00.. 438°7 197'8 79°3 
Tsobuty! butyrate ............... 430°8 199-2 737 
Erapy! valerate, niii..<caci0s 4405 197'8 75°6 
Isopropyl valerate ............ 429 197°2 137 
Amy]! butyrate ................6- 457°8 222°3 756 
Methyl! nonylate ............... 4775 245°7 76°2 
Methyl acrylate .......4....... 353'3 98:4 76°5 
Eeopylacrylate .....c..scc000 395°9 1449 754 
eoraidehyde .......es-ceeceecs 294 56°9 76-4 
Valeraldehyde .................. 374 1199 758 
PaewlGenyde ............c0eesa0et 3974 150°7 14:7 
Aromatic Hydrocarbons. 
1 eee 353°2 95-9 77°3 
co oe ere 3822 1179 179 
BPE occa: cee seseneanues 414-7 138°7 80:1 
EMOTE 5.0. cs ctscaccuseceesaese 412 139°7 794 
=. VL rs 411 140-2 79°2 
ENGHYEDERZENE ... 62.20... ccnceee 409°3 1588 79°1 
DRE fese cen s20ccs-0cevcecsaesen 4483 184-4 738 
Aliphatic Hydrocarbons. 
2 341°9 139:8 65'8 
1 re 371-4 161°8 68:2 
2) ee 331 136°3 64:3 
JUSS. 5d ee 382-2 1845 67:24 
LC re 432°6 Zol's 705 
CE 01 rer 314 104 66°8 
Aliphatic Ethers. 
IEIONOGDER 2... -<..ccnes- 2.000. 3076 1061 64-9 
Methyl propyl ether ......... 3L19 105°1 66° 1 
Methyl! butyl ether ............ 345°3 127-2 63°3 
Ethyl propyl ether ......... wa 336°6 127°8 66°9 
Hitiyh butyl ether  ........5:.. 3864 150°1 63-6 
DMGy Ether ..060.<c0evereeees 4139 197°3 711 
Aromatic Esters. 
Methyl benzoate .............. 463 150°3 871 
Ethyl benzoate ...1.........00.06 482 174:2 86:3 
Methyl phenyl propionate ... 509°6 195°2 87:9 
Ethyl phenyl propionate...... 521°1 2215 8673 
n. Propyl pheny! propionate. . 539°1 2459 85-4 


Phil. Mag. 8. 6. Vol. 20, No. 117. Sept. 1910, 2.N 


330 


Liquid. 


Aromatic thers, 


Phenyl propyl ether.. 
Phenyl butyl ether .. 
Phenyl octyl ether .. 
o-Tolyl ethyl.ether .. 


o-Tolyl butyl ether 
m-Jolyl ethyl ether 
m-Toly] butyl ether 
p-Lolyl propyl ether 


Aliphatie Chlorides. 


Ethyl chloride ........ 
Propyl chloride ..... 


Isopropyl chloride 
| Isoamy! chloride 


Aliphatic Broimides, 


Methyl bromide ..... 
Ethyl bromide ........ 
Propyl bromide ...., 
Isopropyl bromide .. 


Aliphatic Iodides. 


Methyl iodide ........ 
Hithyl iodide............ 
Proapyliodide 2.232. 
n. Amyl iodide ........ 
Butyl iodide -......:.:. 


Amines. 


Diethylamine ........ 
Triethylamine ........ 
Tripropylamine ..... 
Isobutylamine ........ 


Chloroform woeccssecc: 


Carbon tetrachloride 


Hthylene chloride..... 
Ethylidene chloride .. 
Chlorobenzene ........ 
Bromobenzene ........ 
Todobenzene ........... 


Nitric peroxide NO, 


Sulphur dioxide ..... 
Carbon bisulphide .. 
Tin tetrachloride ..... 


p-Lolyl ethyl ether .. 


Amyl chloride ... Rte 
Altyliehtoride ........ 


Temperature 
of Boiling Tsp 
(absolute). 


wee eee 


aes seer ene 


meee ene nee 


eee ewe eee 


Tete wwe aee 


Sete ewe eee 


er ra 


eee eeeeee 


eee ww eeee 


eae e seen 


ee) 


re ror 


eee eeseeee 


eo 2e@eceese- 


463°5 
483°3 
555°8 
457°8 
495:0 
465-0 
502°2 
462°9 
483-4 


285°2 
319-2 
309°5 
372°6 
o7+5 
318°5 


286 
o 1 2-7 
344 
333 


315'8 
345°3 
375°5 
424-7 
4029 


329 
362 
429-5 
340°7 


3341 
349°7 
3596'5 
331°8 
405:0 
429.0 
4614 
2946 
265 

319 

387°1 


Molecular 


Mr. D. Tyrer on Relations between the Physical 


Volume V at |) Bg oeoes 
Boiling-Point. V 
172.0 83°3 
1953 83:3 
2961 83°4 
170-9 82:5 
218-4 82:2 
172-0 83:6 
23-5 heal 
Aa 83:2 
196-0 83°2 
"2 68°8 
0 es TO 
93°6 68:2 
134-4 Ta 
136°3 72-5 
84°5 726 
58-2 73:8 
brig ib 73°0 
. 97-2 74:8 
97°2 71:9 
84:1 783'1 
86:1 78:2 
106°9 79:1 
150-4 79-9 
128°2 79:9 
109°1 68'8 
153°8 67-6 
22-1 “0:9 
1062 71:9 
84:5 761 
103°7 74:3 
85:2 81-0 
88°7 74:4 
114:3 83-5 
119°8 85:0 
130°6 90°9 
63:9 73°7 
43°9 tow 
621 80:6 
131:2 76:3 


Properties of Liquids at the Boiling-Point. d31 


Inquids which show Molecular Association, 


| Temperature Molecular Tp» 
Liquid. _ of Boiling T,, | Volume V at K, = ax 
(absolute). Boiling-Point. 
ALAN) axs5e seis jy cce ea | 373 18:78 140°3 
Methy] alcohol ..............+... 337-9 42-6 971 
Meny! alcool: in. asc.c- seas ; 3313 62-2 88°7 
my Eropy! alcohol »: {2 .0:, e2etees 3704 81-2 85:1 
iy Hexyl aleahal *...).2.ae 429°6 146°3 815 
HiGrMiie AChE Mics. 82 ae 373 ee | 108-1 
PREETI TCH ek ap « cite aceon asian 39L°5 63°6 980 
Propionic acid *!: 5. 14... 4138 85°5 93°9 
BAHL TIC ACI 28.) 652.2 Og. las ae 435 108 91°3 
Methyl formate ............... 304-7 60:10 76°65 
Methyl oxalate ........,....0000 / 435°6 116°8 89-1 
Mithyl oxalate “/....2.0..2...0ee¢4 459-0 166°2 83 5 
Erapy! oxalate. 2..5.%2<00: sees 486°5 215-4 81:2 
Dimethyl succinate ..........., 468-2 159°7 86°3 
Methyl ethyl succinate ..,... 481°2 1846 84-6 
Diethyl succinate ............... 489°2 209-0 82°5 
PR AMNLTILO ) 01). nc wedoaepee 347 54°3 91°7 
EOPIDNILTIG§,- 5.5.00. 0c Jesdeuns | 370 73:3 80:1 
Memernarheile’.)... 002. .c0.se2aees : 402 1107 81:6 
Lt 4 eee * 467 103°6 99°4 


It will be observed from the Table that the relation gives 
an approximately constant value of K for all the homologous 
series, but the value varies considerably from one homologous 
series to another. In general the higher boiling liquids 
seem to give rather high values of K. The approximate 
values of the constant for the different homologous series 
are given below. 


_The aliphatic hydrocarbons and ethers give K=68. 


The aliphatic chlorides and amines give K=70. 

The aliphatic esters and bromides give K=74. 

The aliphatic iodides and aromatic hydrocarbons give K=79. 
The aromatic ethers give K=83. 


The validity of the relation is also considerably affected 
by molecular association, as the latter portion of the above 
Table indicates. For associated liquids the equation should 
obviously be written 


T = K./Vn, 
where n isthe association factor. 


Were it not for the uncertainty of the value of K for 
2N2 


532 Physical Properties of Liquids at the Boiling-Point. 


any particular liquid, the above equation would furnish a 
method of determining n the association factor. 

There is one way of distinguishing between the effects of 
association and constitution. If the members of an homo- 
logous series give a gradually diminishing value of K as 
the series is ascended, then the members of that particular 
series show molecular association. If the value of K remains 
approximately constant then no molecular association occurs. 
For this reason the esters of the aliphatic dibasic acids, like 
-oxalic and succinic, have been classed along with the alcohols 
cand acids as associated liquids. 

Mention must be made here of other relations between the 
molecular volume and the boiling-point. It has long been 
noticed in a general way that there is a certain parallelism 
between the molecular volume and the boiling-point, but no 
exact relation has yet been discovered. Masson (Phil. Mag. 


Vv 
—=constant, where V and T are the molecular volume and 


aL 


temperature of boiling respectively, which he found to hold 
closely for the alkyl halides, but apparently only for these few 
classes of substances. 3 

On the other hand, Young (Phil. Mag. vol, xxx. p. 423, 1890) 
showed that Masson’s relation should only hold for substances 
which have the same critical pressures, and deduced the 


relation (from van der Waals’ generalizations) 
V = const. x e 


where V is the molecular volume at the boiling-point, T, is 
the critical temperature, and P, the critical pressure. 
Obviously for those liquids where P, is constant, Masson’s 
relation will hold. 


Summary. 


An empirical relation between the latent heat of a liquid 
and the molecular volume at the boiling-point is shown to 
hold for most classes of liquids, with the exception of the 
aliphatic hydrocarbons and ethers and liquids which show 
molecular association. The relation may be written 


LM =K V/V, 


where L is the latent heat at the boiling-point, M the mole- 
eular weight, V the molecular volume at the boiling-point, 
and K is a constant equal to 1583. The effect of molecular 


| 
j 


ee 


Eneray Relations of Certain Detectors. 533 


association on the validity of the relation is examined and 
it is shown that with associated liquids the calculated value 
of the latent heat may be either greater or smaller than the 
actual observed value. 

The relation combined with Trouton’s equation gives the 
simple relation 


eK AAV, 


where T is the boiling-point, V the molecular volume at the 
boiling-point, and K is a constant. This relation gives a 
fairly constant value of K for the members of an homologous 
series, but its value varies rather considerably from one 
homologous series to another. The relation is also greatly 
affected by molecular association, the value of K always 
being much greater for associated liquids than the average. 


The Chemical Department, 
The University, Manchester. 


LIL. The Energy Relations of Certain Detectors used in 
Wireless Telegraphy. By W.H. Eccuns, D.Sc., A.R.CS.* 


\ } [Plate X. ] 


HE results of an experimental examination into the 
physical properties of four very different types of detector 
used in radio-telegraphy are set forth briefly in the following 
pages. The conditions of the experiments have been made 
generally identical with those arising in the ordinary employ- 
ment of the detectors, and, in particular, the quantities of 
energy given to the instruments, in the form of electrical 
oscillations, have been of the same order in these experiments 
as in actual practice. The detectors investigated are the 
electrolytic, the carborundum rectifier, the zincite detector, 
and a thermoelectric detector. All these have before this 
been subjected to close scrutiny by various observers, who, 
however, used methods different from that of this paper; a 
summary of their work will be given alongside the results of 
the present experiments. These results, it will be seen, are 
expressed in the form of curves rather than as tables of 
figures; each curve may be regarded as typical of the 
detector concerned, and has been selected from a number 
of curves drawn from measurements accumulated during 
last year. 
The method and apparatus used are the same as were 
described in a paper ‘‘ On Coherers,” read before the Physical 


‘ * Communicated by the Physical Society : read July 8, 1910. 


ee 


034 Dr. W. H. Eccles on the Energy Relations of 
Society in March last (Phil. Mag. June 1910). - The present 


experiments were for the most part carried out at an earlier 
date than those described in that paper. The properties of 
the detectors are examined in three distinct ways. The first 
way consists in applying to the detector an electromotive 
foree which is gradually increased, and measuring the con- 
sequent current at each step. The second way is to fix the 
electromotive force at some particular value, to send trains 
of oscillations of various energy values through the instru- 
ment, and to measure the intensity of sound produced in the 
telephone on each occasion. The third way is to send trains 
of constant energy value through the instrument while the 
steady electromotive force applied to it is varied in steps, 
and to measure at each step the intensity of the sound pro- 
duced in the telephone. These modes of experimenting give 
curves that may be called respectively the steady-current 
curve, the power curve, and the sensitiveness curve. It will 
be seen from the curves that the power supplied to the de- 
tector in the form of electrical oscillations and the power 
handed to the telephone in the form of intermittent current 
are both recorded in fractions of a watt. For this purpose, 
the circuit-calibrations described in the former paper were 
used. It must be mentioned here that the calibration of the 
telephone circuit is probably much less accurate than that of 
the detector circuit. 


RESULTS OF} EE MEASUREMENTS. 
The Electrolytic Detector. 


This detector consisted of two platinum electrodes in dilute 
sulphuric acid (one of acid to four of water). One electrode 
was a platinum wire of 0-0006 cm. diameter drawn by the 
Wollaston process, dipping a fraction of a millimetre into 
the electrolyte ; the other was a piece of thick wire well 
immersed. When a potential difference less than one volt is 
established between the electrodes the current that passes 
is very small ; but as the potential difference is increased the 
counter electromotive force of polarization is overcome, till 
finally a large current flows with evolution of gas. The 
stage of the process which is useful for detecting feeble 
electrical oscillations is that where the bubbles of gas do not 
yet form on and break away from the point freely. The 
steady current curves of fig. 1 (PI. X.) indicate the difference 
between the two cases, point as anode and point as cathode. 
The power curves for various cases are collected in fig. 2, all 
from the same detector. Curve a shows the relation between 


Certain Detectors used in Wireless Telegraphy. 435 


the power w delivered by the detector to the telephone and the 
power W given in the form of electrical oscillations to the 
detector, when the potential difference between the platinum 
point and the large electrode has the value 2°9 volts—the 
voltage of highest sensitiveness. Curve db shows the large 
fall in sensitiveness caused by altering the applied electro- 
motive force, the point still being anode. 

In the same way, curves ¢ show the efficiency of the energy 
transformation when the point is negative. Curve ¢ is an 
attempt to reach the best possible sensitiveness with the 
point as cathode ; but it is to be remarked that when the 
point is covered with hydrogen, the electrical conditions are 
somewhat unstable, and the best potential difference is an 
uncertain quantity. This is in strong contrast with the very 
definite conditions that rule when the platinum point is 
polarized with oxygen. This is clearly indicated by the 
curves of fig. 3, where the ordinates represent the proportion 
of energy delivered to the telephone when the applied voltage 
has various values. 

This detector has received a very great deal of attention in 
the past, but the precise mode of operation of the instrument 
is still unknown. Reich*, after making experiments with 
superposed direct current and alternating current of low 
frequency, considered that the phenomena could only be 
accounted for by a dissolution of the small electrode used as 
anode and a simultaneous disappearance of oxygen—pro- 
cesses purely chemical. Rothmund and Lessing], using a 
Blondlot oscillator and Lecher wires, and measuring the 
potential difference across the detector and the current 
through it when the oscillator was working and not working, 
concluded that the whole action of the instrument depended 
upon some unexplained “ depolarization action” of the oscil- 
lations ; and by using a variety of electrolytes, proved the 
incorrectness of the hypothesis, which was a mere surmise 
unsupported by scientitic measurements, that the instrument 
operated by resistance alterations due to the heat generated 
in the liquid mass near the minute anode. Later Dieckmannt 
measured the current changes that followed upon the passage 
of strong oscillations of (unmeasured) intensity through a 
detector. Later still, Austin §, using alternating current of 
low frequency, showed that the detector was affected by 


* Phys. Zettschr. vy. p. 838 (1904). 

+ Ann. d. Phys. xv. 1, p..193 (1904). 

{ Phys. Zeitschr. v. p. 529 (1904). 

§ Bulletin, Bureau of Standards, i. 3, p. 435 (1905). 


536 = =Dr. W. H. Eccles on the Energy Relations of 


electromotive amplitudes of 1/10000 volt, and that oscilla- 
tions produced by spark-discharges in the laboratory affected 
the detector equally whether the small electrode was anode 
or cathode. He concluded that in the action of the instru- 
ment heat had a share, and that chemical action, electrostatic 
attraction across the gas film, and also a property styled 
rectification, all took part. 

Carborundum.—The detector was set up by clamping a 
erystal of carborundum between brass plates, so that a smooth 
crystalline edge or corner was in contact with one plate, and 
a blunt and more amorphous part of the crystal in contact 
with the other plate. The steady current curves of two 
crystals widely different in their electrical behaviour appear 
in fig.4(Pl. X.). Curves ab belong to one crystal, the dotted 
curves AB belong to another. The upper curve of each pair 
was obtained when the jagged blunt end of the crystal was 
positive. 

The power curves are given in fig. 5. Lines abe exhibit 
the energy relations for the crystal that gave ab in fig. 4. 
Line a was obtained while the blunt end was at a potential 
2°62 volt higher than the smooth end; 6 was got when the 
potential difference was —0:44; curve ¢ was got without 
electromotive force. Lines A, B,C refer to the other crystal. 
Line A was obtained while the blunt end was 2°9 volt above the 
smooth end; line B while the blunt end was 2°1 volt below ; 
line C while no external electromotive force was applied. 
The curve of fig. 6 shows how sensitiveness altered with the 
electromotive force applied to the terminals of the detector. 

The Carborundum detector has been examined very ex- 
haustively by Pierce*. He has shown that crystals of this 
substance may be as much as 1000 times more conductive 
for current in one direction than in the opposite, and has 
concluded that the substance acts as a detector of high 
frequency oscillations solely because of this unilateral con- 
ductivity. Heat, he considered, played no part in the process. 
The curves given above show, however, that a crystal may 
be a good detector even though its unilateral conductivity 
be not very pronounced. 

Zincite-chaleopyrite.—The detector made by arranging a 
corner of a fragment of brown zincite (native oxide of zinc) 
to press against a piece of chalcopyrite (iron copper sulphide) 
is one of the most sensitive known. It is used extensively 
in various navies. Tig. 7 gives the results of measurements 
of current under steady electromotive force. For this par- 
ticular detector the most sensitive condition was attained 


* Phys. Review, xxv. p. 31 (1907). 


Certain Detectors used in Wireless Telegraphy. 537 


when the zincite was maintained at a potential about 0°45 
volt below that of the pyrite. The power curves are given 
in fig. 8; here a is the curve when no electromotive force 
was applied: 4 is the curve for an applied electromotive 
force of 0°45 volt, zincite negative, and ¢ is the curve for 
an applied electromotive force of 0°45 volt, zincite positive. 
It will be seen from these that the combination forms a very 
sensitive detector, even when no external electromotive force 
is applied. The curves showing the change in sensitiveness 
with variation of the applied electromotive force are plotted 
in fig. 9. 

Graphite-Galena.——A detector that is very widely used for 
every-day telegraphy is that consisting of a pointed piece of 
graphite touching the face of a crvstalof galena. The curves 
connecting applied electromotive force and current flowing 
through the contact appear in fig. 10. The power curves 
are shown in fig. 11: curve a is obtained when the external 
electromotive force is not applied, and curves 6 and ¢ when 
electromotive forces of 0°45 volt and —0°45 volt were applied. 
The connexion between the power given to the telephone 
and the electromotive force applied to the detector is given 
in fig. 12. 

These two last detectors and others similar to them are 
sometimes called ‘“ rectifiers,’ sometimes ‘thermoelectric 
detectors.” They are styled thermoelectric because it was 
originally supposed that they owed their power of detecting 
high frequency vibrations to the thermoelectremotive forces 
set up at the contact by the rise of temperature produced at 
that point—the point of highest resistance in the whole 
oscillation circuit—in obedience to Joule’s law; but most 
observers have concluded from experiments with both direct 
and alternating currents, that these detectors derive their 
function from an unexplained and hitherto unknown power 
of rectifying rather than from a combination of the Joule 
and Peltier effects. Pierce* has examined the behaviour of 
contacts made with anastase, brookite, and molybdenite under 
alternating currents of ordinary frequency, and obtained 
oscillograms of the current through them. No evidence of 
thermoelectric or other integrative action was perceived in 
the photographs. Austint has examined quantitatively, also 
by aid of slow alternating currents, the properties of detectors 
consisting of contacts of silicon and steel, carbon and steel, 
tellurium and aluminium. Brandes{ and Raetenkrantz § 

* Phys. Review, p. 153 (1909). 
¥* Bulletin, Bureau of Standards, v. p. 183 (1908). 

y Llektro. Zeitsehr. xxvii. p. 1015 (1906), 

§ Phys. Zeitschr. ix. p. 911 (1908). 


538 Mr. G. W. de Tunzelmann: Jflechanical Pressure of 


have also contributed greatly to our knowledge of the con- 
nexion between the steady current curve of a detector and 
its behaviour under electrical oscillations. 


Conclusion. 


The chief fact brought to light by the above experiments 
is that the energy passed to the telephone by a detector is 
connected linearly with the energy given to the detector in 
the form of electrical oscillations. “This is true for all the 
detectors examined, even including the coherers discussed in 
the earlier paper. ‘The curves connecting the input and 
output of energy though they are straight lines usually 
pass some distance away from the origin. ‘This implies that 
for a particular detector under invariable conditions there is 
a fixed wastage of oscillation energy, amounting commonly 
to about 1/10 of an erg per second, however large or small 
the oscillation energy given to the detector may be. Another 
interpretation is, howev er, that a small quantity of ener ey, 
wiichas Gnvariable while the detector) 1s undisturbed, 
delivered by the detector to the telephone circuit in a for 
that never makes any proportion of itself manifest as sound. 
The curves suggest, though they do not prove, that all 
detectors are fundamentally thermal in their action. That 
this deduction is opposed to the conclusions reached hy 
previous experimenters is clear from the summary of their 
work given above. The principal cause of this difference 
between our conclusions appears to be that nearly all previous 
observers have used comparatively large quantities of oscil- 
Jation energy, and have therefore probably brought into 
play phenomena that never arise in detectors as used in 
wireless telegraphy. 

The above investigations were carried out by the aid 
of a grant from the Royal Society’s Government Grant 
Committee. 


LIV. The Mechanical Pressure of Fadiation effective on 
the smallest as well as on larger Particles. By G. W. 
DE TUNZELMANN, B.Sc.* 


TV has been pointed out by Prof. Schuster in a letter to 

‘Nature’ + that “there is a widespread impression that 
light pressure acts only on particles the linear dimensions of 
which include several wave-lengths of light, but this is not 


* : Ea by the Author. 
‘Nature,’ vol, lxxxi, p. 97 (1909). 


Radiation effective on smallest as well as larger Particles. 539 


correct. The determining factor is the extinction of light, 
whether by scattering or by absorption, as indeed appears if 
we take the view adopted in Prof. Poynting’s work on the 
subject, that a propagation of momentum accompanies the 
transmission of light. The momentum is destroyed equally 
whether the molecules act as scattering or absorbing 
centres.” 

This conclusion is in accordance with Tyndall’s experi- 
ments* on the colours of precipitated clouds of small 
particles, and on the blue colour and polarization of the light 
from the sky, and Lord Rayleigh’s theoretical investigations 
suggested by themt; and with the experimental evidence, 
adduced by Lord Rayleigh in the last paper referred to, 
tending to the conclusion that the molecules of air are 
responsible for nearly a third of the atmospheric scattering 
observed. 

The cases of plane waves normally incident upon a per- 
fectly black body and a perfect reflector are capable of 
simple treatment by elementary mathematical methods ft, and 
since the work done across a small surface of a wave-front 
cannot depend on the question whether the wave is plane or 
not, the relation so arrived at must hold good for any simply 
periodic electromagnetic disturbance. The subject is treated 
on these lines in a recent work by the present writer §, and 
a more general analytical investigation is given by Sir 
Joseph Larmor on p. 131 of ‘ Aither and Matter.’ It is 
shown in chapters vii. and viii. of the latter work that 
Maxwell’s equations of electric force are not applicable to 
the investigation of problems in which radiation is important. 
These equations are derived from an electrodynamic stress- 
formula in which the function of a uniform dielectric is 
regarded as merely to transmit the forces without adding 
anything to them. In the light of present knowledge, which 
is most completely formulated in terms of the electron theory, 
any material dielectric must be regarded as susceptible of 
polarization analogous to that of a magnet. Now, while in 
metallic conduction the current arising trom this polarization 
is usually negligible in comparison with the total, in radiation 
it forms an important part of the total current. 


* Phil. Mag. vol. xxxvii. 1869, p. 884; and Phil. Trans, vol. clx. 
1870, p. 388. . 

+ Scientific Papers, vol. i. pp. 87, 104, 518; and vol. iv. pp. 305, 397. 

t See Drude’s Lehrbuch der Optik, p. 447, or English edition, p. 488, 
and Sir Joseph Larmor’s article “ Radiation,” Supplen ent to Encyclopedia 
Britannica, 

§ Treatise on Electrical Theory and the Problem of the Universe, 
pp. 270-274, 


540 Notices respecting New Books. 


A detailed theory of radiation pressure was developed by 
K. Schwarzschild*, and published in 1901, in which, by 
means of somewhat intricate mathematical analysis, he suc- 
ceeded in arriving at numerical results, according to which 
the radiation pressure on a spherical particle vanishes when 
the radius is too small to include several wave-lengths of the 
incident radiation. The investigation is, however, based on 
Maxwell’s equations of electric force, in which the existence 
of the polarization current is not recognized, and is therefore 
invalid. 

Dr. J. W. Nicholson, in his recent paper ‘On the Size of 
the Tail-particles of Comets, and their Scattering Hffect on 
Sunlight ’’t, relies largely on Schwarzschild’s results. The 
writer was also referred to these results in December 1909, 
by Prof. Svanté Arrhenius, when kindly reading some of the 
proofs of the work previously referred to, in which some 
mcdifications had been made in the theory proposed by Prof. 
Arrhenius to account for the constitution of the solar corona 
and the origin of the polar auroras. It therefore appeared 
advisable to draw attention to their unsoundness in a more 
prominent way than has already been done in a footnote to 
the writer’s recent work f. 


LV. Notices respecting New Books. 


Mathematical and Physical Papers. Vol. 1V. Hydrodynamics and 
General Dynamics. By the Right Honourable Sir WILLIAM 
Tuomson, Baron Kervin. Arranged and Revised with briet 
annotations by Sir Josepu Larmor, D.Sc., LL.D., Sec. B.S. 
Cambridge: at the University Press. 1910. 


| Poe ENTY years have elapsed since the Third Volume of Kelvin’s 

collected papers was issued. During that period Kelvin 
himself edited and greatly expanded the Baltimore Lectures, and 
added many new papers to the already long list of original contri- 
butions to science. But the remarkable papers on Vortex Motion 
remained generally inaccessible save to students in command of a 
good scientific library ; and many had to be content with the pre- 
sentation of Kelvin’s work as it was given in the Treatises of 
Lamb and Basset. Now, thanks to the care and energy of 
Sir Joseph Larmor, this has been changed. The present volume 


* Muinchener Berichte, vol. xxxi. 1991, p. 298. 
T Phil. Mag. vol. xix, 1910, p. 626. 
i Op. cit. p. 378. 


Notices respecting New Books. 541 


opens with the short paper on Vortex Atoms, which was commu- 
nicated to the Royal Society of Edinburgh in 1867, and was 
published both in the Proceedings of that Society and in the 
Philosophical Magazine. Then follows the great memoir ‘ On 
Vortex Motion,” published in the Transactions of the Royal 
Society of Edinburgh, vol. xxv., which has been for many years 
accessible only in the older scientific libraries of the world. 
Probably the vast majority of students of hydrodynamics have 
never had a good opportunity of reading this great paper. No 
doubt the outstanding features of Kelvin’s mode of presentation — 
especially’ the conceptions of Flow and Circulation—are well 
given in our standard treatises on the motion of fluids; but Kelvin 
had a method all his own, full of suggestiveness to the thoughtful 
reader. It is of infinite value to the real student to have ready 
access to the original work of a man like Kelvin, especially when, 
as in the present instance, each series of papers forms a kind of 
continuity. The arrangement is broadly by subject matter ; and 
in each section a chronological arrangement is made the basis. 
Thus the Hydrodynamic section includes the papers on vortex 
motion, on the motion of solids through fluids, and on capillary 
waves. Then come three papers on the Tides; and under the 
heading Waves on Water are grouped a number of connected 
investigations on stationary waves, ship waves, the front and rear 
of a free procession of waves, and so on. Several of these papers 
deal with difficult subjects, and constitute some of the latest of 
Kelvin’s most characteristic contributions to the theory of certain 
types of water waves. The five papers communicated to the 
Royal Society of Edinburgh at intervals from 1904 to 1906 form a 
continuous series, the paragraphs being numbered consecutively. 
These portions occupy about four-fifths of the volume. The 
remaining fifth is concerned with General Dynamics and Elastic 
Propagation ; but many of the papers enumerated in the Table of 
Contents are represented only by their titles, the papers having 
being already reprinted in the Baltimore Lectures, or in the 
earlier volumes of ‘ Mathematical and Physical Papers.’ The last 
paper printed in extenso is on anew method for specifying stress 
and strain in an elastic solid. In place of the usual Cartesian 
specification, Kelvin uses a tetrahedron of reference, and is thus 
able to obtain a symmetrical specification of stress and strain for 
finite as well as for infinitesimal strains. It need hardly be said 
that the editorial work has been well and faithfully done, the 
annotations, brief though they are, being always to the point and 
full of instructive allusions. The final volume (V.), we are told, 
is almost ready for press, and will contain papers on Thermo- 
dynamics, Cosmical and Geological Physics, Electrodynamics and 
Elee‘rolysis, Molecular and Crystalline Theory, eesloncUiyiliny and 
Electrionic Theory, and other miscellaneous matter. 


O42 Notices respecting New Books. 


An Elementary Treatment of the Theory of Spinning Tops and Gyro- 
scope Motion. By H. Crasrren. Longmans, Green & Co, 
London: 1909. 

Tue history of applied mathematics is full of illustrations of 
the truth that problems which tried the powers of the most 
powerful thinker of one age become the familiar possession of the 
average student in the succeeding generation. Usually the first 
presentation of the theory is crude and difficult ; but with the 
flight of the years the combined attacks of many minds evolve a 
simpler and generally a more natural way of looking at the asso- 
ciated problems, which finally find their appropriate place in a 
so-called elementary textbook. Spinning tops and gyroscopic 
motion are a case in point. In this refreshing little book which 
Mr. Crabtree has prepared, the dynamical principles of the con- 
servation of energy and the conservation of moment of momentum 
are applied with simplicity and power to many familiar phe- 
nomena of motion—spinning tops, mono-rail, precession, diabolo, 
torpedo, golf-ball, rifle-bullet, and so on. Simple diagrams help 
to elucidate the discussion—with the exception (if a fault must be 
noted) of the misleading diagram on page 53 of the flight of a 
rifle-bullet. Here also the attempt to explain the “ drift” is not 
convincing—indeed, the hydrokinetics is not sound. ‘lhe foot- 
note on the same page referring to the golf-ball and Tait’s investi- 
gations is so full of misstatement and the mythology of the Club- 
house that the author had better suppress it altogether in a second 
edition. These tlaws apart, the book deals in an interesting 
way with great problems. There are many examples for the 
student to work at; the mathematics is kept well under control, 
and the dynamical ideas are never lost sight of. The book should 
be in the hands of all students who are beginning their higher 
studies in applied mathematics. 


Funktionentafeln mit Formeln und Kurven. Von Dr. E. JAHNKE 
und Ingenieur Ff, Empse. Teubner: Leipzig und Berlin. 1909. 


THeEsr tables of functions of various kinds are a most valuable 
addition to the many books of tables which have recently been 
prepared. The mere enumeration of some of the functions which 
are here tabulated will suffice to show to all interested the real 
importance of the work: Exponential Functions, Hyperbolic 
Functions, Fresnel’s Integrals, the Gamma Function, the Error 
Integral, Elliptic Integrals and Functions, Spherical Harmonics, 
Bessel’s Functions, ete. Among the elliptic. functions there are 
not onlv the familiar functions in their earliest form, but also 
Theta functions, and the Weierstrass functions; and a valuable 
addition gives the mutual induction and attraction between 
parallel co-axial circular circuits in terms of a quantity which 
depends on the radii of the circuits and their distance apart. © 


Geological Society. 43 


Here, as in other cases, the tables are supplemented by the corre- 
sponding curves or graphs. This graphical representation is 
indeed an extremely important feature throughout. Hach Table 
is accompanied by an exposition of the theory sufficient to refresh 
the memory of the worker in applied inathematics who wishes to 
make use of it. In most cases the values are given to four or five 
significant figures. The names of the authors are a guarantee of 
the care that must have been taken in preparing this most timely 
publication. 


LVI. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from vol. xix. p. 918. ] 


January 26th, 1910.—Prof. W. J. Sollas, LL.D., Sc.D., F.R.S., 
President, in the Chair. 


ies following communications were read :— 


1. ‘On a Skull of Megalosaurus from the Great Oolite of 
Minchinhampton. By Arthur Smith Woodward, LL.D., F.RS., 
F.L.S., Sec. G.S. 


2. ‘Problems of Ore-Deposition in the Lead and Zinc Veins of 
Great Britain.’ By Alexander Moncrieff Finlayson, M.Sc., F.G.S. 

Chemical analyses show traces of lead and zinc in several of the 
rock-formations of Britain, but the ores of the veins are concluded 
to be derived, not from the country-rock, but from deeper sources, 
probably in the first place by magmatic segregation. They were 
transported in the deeper zones by ‘juvenile’ waters, in which 
fluorine was an important constituent, while in the upper zones, 
especially in limestone districts, underground waters of meteoric 
origin have played a large part. The vein-solutions carried (1) 
alkaline sulphides, which held the sulphides of the metals in 
solution, and (2) alkaline and earthy carbonates, The presence 
of the latter is indicated by the alteration of the wall-rock, which 
shows aconcentration of potash, lime, and carbon dioxide, and a 
leeching of soda, magnesia, oxides of iron, and silica. In lime- 
stones, however, the -chief effects of solution on wall-rock were 
concentration of silica and magnesia. 

The filling of fissures rather than direct replacement of rocks by 
ores, has been the chief process, but the calcium of fluorspar has 
been very largely derived from the country-rock. Further, much 
local metasomatism is seen, such as replacement of limestone by 
fluorspar, galena, blende, and quartz; and replacement of fluorspar 
by galena. 

The order of deposition, determined by microscopic examination 
of polished specimens of ores, has been: chalcopyrite, fluorspar, 


544 Intelligence and Miscellaneous Articles. 


blende, galena. The galena carries its silver generally in molecular 
or isomorphous combinations, except in the case of rich ores, when 
native silver and argentite appear sometimes as threads along the 
cleavage-planes. 

In the effect of the country-rock on ore-deposition, the chief 
factors have been: (1) the physical character of the rock and the 
consequent natnre of the fissure, (2) its porosity, and (3) its chemical 
composition. The process of deposition involves interchange of 
constituents between rock and solutions, even with the least soluble 


‘rocks. 


Ore-deposition has persisted over a vertical range of 5000 -to 
6000 feet, of which over one-half has been shorn off by denudation. 
The effects of secondary processes have been exerted to depths of 
over 600 feet. The main points in the work are supported by 
field-observations, and by the results of microscopic and chemical 
research. 


3. ‘The Vertebrate Fauna found in the Cave-Earth at Dog Holes, 
Warton Crag (Lancashire).’ By John Wilfrid Jackson, F.G.S., 
Assistant Keeper in the Manchester Museum. 


LVIL. Intelligence and Miscellaneous Articles. 


To the Kditors of the Philosophical Magazine. 


ice aes Cavendish Laboratory, Cambridge, 
* ? Aug. 9th, 1910. 

W ITH reference to the paper in your last issue by Mr. Jeans 

on the motion of an electrified particle near an electrical 
doublet, and its bearing on the theory of radiation given by me in the 
June number, I should like to state that I was quite aware that the 
state of steady motion I considered would not be permanent. As 
a matter of fact I showed in the paper that if the particle suffered 
a radial displacement from its circular orbit, no force acted upon 
it tending to bring it back to its former position or to drive it 
still further away; thus if the particle were started with a radial 
velocity it would slowly drift from its state of steady motion. 
I did not then, nor do I now, consider this fact of any import- 
ance with respect to thetheory of radiation I was discussing, 
for there is nothing in that theory which requires these systems to 
be permanent. All that is necessary is that, at any time, there 
should be a number (infinitesimal in comparison with the number 
of molecules) of such systems which remain in this state, or only 
depart slightly from it, in the time occupied by a few vibrations of 
ultra-violet lght. | 

Yours very sincerely, 
J. J. THOMsoN. 


= ———— ay. eS SS CN IC NTA 3 TLL le ie eee ne een sale eS - 


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Phil. Mag. Ser. 6, Vol. 20, Pl. VI. 
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Phil. Mag. Ser. 6, Vol. 20, Pl. VI. 


Fie. 14. 


BARTON & EBBLEWGITE. 


Fie. 1.—Optical Arrangements for Longitudinal Motions of Bridge. 


Fig. 3.—Enlarged detail of Bridge and Optical Lever. 


Phil. Mag. Ser. 6, Vol. 20, Pl. VII. 


Fic. 2.—Perspective View of Experimental Arrangements. i 


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a a = x Curves. yi 
7 | Fig. 11.—Powe Snes 8 Watt. 
is) SEO 0-6 os 10 i 14 16 18 Volt 


Fie. 10.—Steady Current Curves. 


= 
1000 | T | 


Mag. Ser. 6, Vol. 20, Pl. X. 


ilk 


Ph 


ELECTROLYTIC DETECTOR. 


+ Pojnt Posi; 


© Po; 


T 
ive 


it Negative 


ha Saale ele [ | 


+ Point Positive 
“9 Volt 
aS + 


+ a 
b: 212 Volt 
| {0} Faint Negative 


C:+ 1-1 Volt 


0 


Fie, 1.—Steady Current Curves, 


3 4 5 6 7 8 9 
107° Wat: 


Fie. 2.—Power Curves. 


Ld 


1 2 3 4 Volt, 
Point Positive 


Fig. 3.—Sensitiveness Curve. 
W=6:2x1075 Watt. 


2 ul 
Point Negative 


ee i. oe 
ECCLES. 


-»* 


2 0-4 


S 
— 


O6 os 10 12 14 16 


Fie. 10.—Steady Current Curves. 


18 Volt 


10°6 Ampere CARBORUNDUM DETECTOR. 
40) 400-—— ——____-___ = 7 2 
el. Ualacine , | 
AB Sra hes e 6 r “f 5§}—_— Tae 
+ Blunt end Positive Wi ° ° | 
fo} Blobt end Negative Af | 
30] 300) + 5 fh | | ee | e/a | ent | BS 
VA 
4 + 44— as 
Dilley ee Hs 
re cu 5 | > S 
20) 200) B 3 - A 2+ _| {————_}__{ __*« | 
| 
es ar 
4 7 3 2 1 0 1 2 Sara asaVell 
Blunt end Negative Bluntend Positive 
10} 100 AE - Fig. 6..—Sensitiyeness Curves. 
[ W =7'85 x 10~* Watt. 
() 1 
e b 
i 5 20" 0 
Volt 
Fie, 4.—Steady Current} Curves, 
ZINCITE-CHALCOPYRITE DETECTOR. 
26 
5 S a ala 
§ — - Sahel + __| + 7 
% 
160 5 +-— | 
H | 
: + incite Positive] (+ 0:45 Volt) 
© Zincite Negative (— 0-45 Volt) 
200 + * NoE.M.F 
| ¢ 
lear 10 1:0 Volt 
+ Zincite Positive Zincite Negative Zincite Positive 
© incite egative OVO 
50) as —— iE 7 Fie, 9.—Sensitiveness Curve. 
10-8 Watt W=62 1078 Watt. 
ules | Fic. 8.—Power Curves. 
ONSNC2 G04 Oban Olas a0 
Volt 
Fic, 7,—Steady Current 
Curves. 
GRAPHITE-GALENA DETECTOR. 
z° Graphith Positive (0-29 Volt 
B § Grad Negative Coes en 
2000] + on + NOEMF. 
Oo ®, . A 
+ Graphite Positive [ 
e Glaphite Renatice 
4 : + 1 
© 
4 2 | 
1000) alt ! : ; 
OSs ek © AO a Ser ea 
Fre. 11.—Power Curves. Fic. 12.—Sensitiveness Curve, 


W =7'85x 107-5 Watt. 


hae 


7 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 
OCTOBER 1910. 


LVIII. The Specific Charge of the Ions emitted by Flot Bodies. 
—II. By O. W. Ricwarpson, W.A., D.Sc., Professor of 
Physics, and K, R. Hunsirr, A.M., Princeton University*. 


+ ae a previous communication} one of the authors deve- 

loped a method of measuring the value of e/m for the 
ions emitted by hot bodies and applied it to the case of the 
ions of both signs from platinum and carbon. As was to be 
expected from the results of earlier investigations, the value 
of the specific charge for the negative ions was found to 
correspond to that tor the negative electrons, whereas the 
number obtained for the positive ions pointed to bodies of 
atomic magnitude. Somewhat contrary to expectation, the 
value of e/m for the positive ions was found to be identical 
within the limits of experimental error for both platinum and 
carbon, despite their wide divergence of atomic weight as 
well as both chemical and physical properties. The values 
found were also very close to the value given by Sir J. J. 
Thomson for the corresponding quantity in the case of iron, 
which appears to have been the only substance for whose 
positive ions the value of e/m had been determined. The 
value of e/m in all these cases was about 380 .M. units, 
and corresponds to an atomic weight of the carriers of about 
26, assuming that they carry the same charge as the hydrogen 
atom in electrolysis. 


* Communicated by the Authors. 
+ O. W. Richardson, Phil. Mag. (6} vol. xvi. p. 740 (1908). 


Phil. Mag. 8. 6. Vol. 20. No. 118. Oct. 1910. 20 


546 Prof. Richardson and Mr. Hulbirt on the 


In view of the unexpected identity of the specific charge 
of the positive ions from these three very different elements, 
it seemed desirable that a greater variety of elements should 
be examined. The present investigation was undertaken 
with that object in yiew, and the materials which have been 
experimented upon include platinum, palladium, gold, silver, 
copper, nickel, iron, osmium, tantalum, tungsten, brass, 
nichrome, and steel. In addition, unsuccessful experiments 
were made upon aluminium, magnesium, and zine. It was 
found that the strips of these metals which were used in- 
variably melted before sufficient ionization could be obtained 
to carry out the necessary measurements. 

The foil from which the strips were cut varied in thickness 
from ‘002 to ‘005 cm., depending on the material. Each 
strip was cut as narrow as possible, none being more than 
"02 cm. wide. 

The platinum, palladium, gold, and silver were cut from pure 
foil supplied by Messrs. Johnson, Matthey & Co. The copper 
was rolled from ordinary magnet wire, and the tantalum was 
rolled from a filament out of a tantalum lamp. The source 
of the other materials, where it is of interest or importance, 
is stated in the context. 

Both the apparatus and the method of using it are precisely 
the same as before. No change was made, even in detail, so 
that for the description of the method it will be sufficient to 
refer to the previous paper. We shall, therefore, content 
ourselves with stating the results which have been obtained. 


Platinum. 


A number of new measurements of e/m have been made 
for the positive ions from platinum, partly to compare with 
the results obtained previously, and partly in order to test 
the working of the apparatus from time to time. Different 
values have been used both of the magnetic intensity and 
the electrostatic potential difference. The results of the 
measurements are given in the following table (p. 547). 

The quantity m/H represents the value of the ratio of the 
mass of the positive ions to that of an atom of hydrogen on 
the assumption that the charge on these ions is the same as 
that carried by an atom of hydrogen in electrolysis. The 
value of e/m for the hydrogen atom has been taken to be 
9°66 x 10° E.M. units. 

It was pointed out in the previous paper that the absolute 
values of e/m obtained by this method were subject to a 
possible uncertainty arising from the theoretical conditions not 


Specijic Charge of the Ions emitted by Hot Bodies. 547 


Platinum, Positive Ions. 


| 
Distance Magnetic Deflexion ef | 
between Intensity H | Volts V. of Ions x (EM ks ts) | 
plates z (cms.). | (lines per cm.). | (1:0635 em.). “one aaa 
“452 4850 82 1°35 275 
‘452 4850 158 "95 262 | 
"452 4850) 320 65 250 
452 4475 | 200 "15 240 | 
"452 4200 82 1:00 202 
"452 3175 83 "885 273 
‘473 4850 119 1:10 222 
‘473 4850 / 80 1°37 233 
‘473 4850 147 1:00 227 


Average values ... e/m=243, m/H=39°8. 


being completely satisfied. It was pointed out that a check 
on this uncertainty could be obtained by determining the 
value of e/m for the negative electrons which are emitted at 
somewhat higher temperatures. The value of this quantity 
is known to be very near 1°88 x 10’, so that a determination 
of the experimental value will enable us to determine the 
correction factor which has to be applied in order to get the 
true value for the positive ions. In making this correction 
we assume that the factor is the same for the positive as for 
the negative ions. There seems to be no reason why this 
should not be very nearly the case. 

The results of the measurements for the negative electrons 
from hot platinum are given in the following table. The 
units in this and all succeeding tables are the same as in the 
preceding table except where the contrary is stated. 


Platinum. Negative Ions. 


Zz. iy ie us e/m. | 


443 28°75 - 74 1°65 Fiax1¢" 
443 28°75 164 1:10 113 
443 14:2 | 41 "95 86 
443 14°2 4] 1:00 “96 
473 142 41 1:225 Fit 
473 14:2 81 975 1:38 
472 14-2 123 Via I-38 
| -473 14-2 41 1-40 1-44 
| -473 142 41 1-33 1:30 
| j t 
Average value ... e/m=1:18X10', 


202 


548 Prof. Richardson and Mr. Hulbirt on the 


The average value of e/m for the negative ions thus comes 
out considerably lower than the standard value 1°88 x 10’. 
The ratio of the two is 1°59. Assuming that the measure- 
ments for the positive ions are subject to the same propor- 
tional error, we get for the corrected values for the positive 
thermions from platinum: e/m=387 and m/H=25:0. 

In what follows we shall apply the correction factor 1°59 
which has been obtained for platinum to the other metals 
investigated. This is legitimate, since the dimensions of the 
apparatus were the same in every case, and with most of the 
materials used it was not possible to push the temperatures 
high enough to obtain the negative ionization without melting 
them. 

In the experiments on platinum the average temperature 
during the experiments on the positive ions was about 900° C. 
(and with the negative ions about 1100° C.). On account 
of the falling off of the positive emission with time, it was 
necessary to raise the temperature from time to time when 
the measurements on the positive ions were being made. 
This effect was not so noticeable with the other metals used, 
and in fact in the case of silver the ionization appeared to 
increase with lapse of time. This may be due to the strip 
used becoming thinner owing to sputtering, and so requiring 
a smaller current to heat it. 

The platinum strips used were always boiled in nitric acid 
before testing. The method could not be used with most of 
the other materials as they were soluble in nitric acid. The 
were, however, cleaned with alcohol and distilled water 
instead. 


Palladium, Copper, Silver, and Nickel. 


The numbers given by the above metals are exhibited in 
the following tables, and do not appear to call for special 
discussion. 


Palladium. Positive Ions. 


14 

Z. | Hi. V. is | e/m. 
433° | 4850 83 105; |) 2a 
443 |. 4850 165 gil ae 182 
‘443 | 4850 329 it | 235 
aise |) ASKO. 41 1:60 | 230 


Average values ... e/m=212, Corrected ... e/m=387. 
| i] i ~45°6, m/H= 28°7, 


Specific Charge of the Ions emitted by Hot Bodies. 549 


Copper. Positive Ions. 


Ze H. wy 2. | e/m. 
578 4850 153 150 238 
‘378 «=| «= 4850 | 196 1-275 291 


Average values ... ¢/m=230, Corrected ... ¢/m=366. 
miH= 42, m/H=26'4. 


Silver. Positive Ions. 


oe H. Vv &. e/m. 
562 4850 160 1°35 225 
"622 4850 197 1°40 200 
622 4850 120 1:80 200 
622 4850 158 1:70 235 


Average values ... e/m=215, Corrected ... e/m=342. 
m/H= 45, miH= 28°3. 


Nickel. Positive Ions. 


7 | H. ¥i at e/m. 
586 | 4850 305 1:10 241 
586 | — 4850 225 1375 278 
‘586 =| «= 4850 143 1525 218 
586 4850 183 1°35 218 


Average values ... e/m=239, Corrected ... e/m=880. 
-m/H= 40-4, m/H= 25°4, 


Osmium. 


This metal was not obtainable in the form of foil or strip, 
so that a filament such as is used in an osmium lamp was 
employed. The principal effect of using a narrow filament 
instead of a strip is to broaden out the pattern in the diagram 
showing the distribution of the ions, so that it is less easy to 
determine the position of the maximum point. The following 


550 Prof. Richardson and Mr. Hulbirt on the 


gives the results of the only measurement which was made 
with the substance. 


Osmium. Positive Ions. 


a H. Vie io e/m. 


587 4850 4] 3°15 264 


Values ... e/m=264, Corrected ... e/m=420. 
mH= 366, m/H= 23:0, 


Gold. 


Gold was found to behave differently from any of the 
preceding metals in so far as the curves obtained when the 
metal was first heated were irregular in outline and very 
broad. The maxima were not as definite as in the preceding 
cases, and the displacement corresponded to a smaller value 
of elm. The character of these curves is well shown by the 
curve with points thus :— x in fig. 1. 


Bip, iL. 


He TS) 


watt TRARY UN. 


(A 


HACE Be 


PASSING THROUGH SLIT 
On 


EAN, 
Vasne RSS 
aS 


l 2 13 
DISPLACEMENT OF SLIT X (l=: 0635 cr.) 


After the gold had been heated for some time the curves 
assumed a more normal shape and at the same time the 
distance between the maxima increased. The curves with 
the points marked thus :—© in fig. 1, which were obtained 


Specyic Charge of the Ions emitted by Hot Bodies. 551 


after the gold had been heated for some time, show those 
characteristics. 
The numbers which are given by two different gold strips 


when they were first heated are shown in the following 
table :—- , 


Gold (freshly heated). Positive Ions. 


z. H. Vi. ze e/m. 
583 4850 160 1-175 tay os) 
583 4850 200 1-00 135 
583 4850 81 1375 103 
487 4650 200 75 168 


Average values ... e/m=139,  Oorrected ... e/m=221. 
m/H= 69°5, mH= 43°7. 


The more definite values given by the more normal curves 
obtained after the metal had been heated for some time are 
shown in the next table. The numbers represent measure- 
ments made on three separate strips. 


Gold (after heating for some time). Positive Ions. 


ee 1st: MW bp e/m. 
563 4850 122 1:85 319 
By G3 4850 162 1:50 eau ueet) 
| +563 4850 202 1:275 254 
583 4850 325 1:10 ie 264 
‘487 4650 200 ‘97 280 


Average values ... e/m=280, Corrected ... ¢/m=445. 
mAh 345, m/H= 217. 


The most natural interpretation of the peculiar behaviour 
of gold would seem to be that the low values of e/m obtained 
from a new wire are caused by the presence of impurities 
which ate very readily volatilized. The values obtained after 
the metal had been heated for some time are not very different 
from those given by the metals which have already been 
examined. It is curious that the initial value of e/m is 
exactly one-half that found later. 


552 Prof. Richardson and Mr. Hulbirt on the 


Iron. 


Very erratic results were obtained with this metal and the 
values of e/m obtained showed no tendency, so far as we have 
been able to observe, to become constant either with lapse of 
time or with any other conditions. This is probably to be 
attributed to irregularities in the emission itself, as it was 
very frequently noticed that during the experiments with 
iron the electrometer spot did not move uniformly, but 
was liable to jerks as though the thermionic emission was 
an intermittent phenomenon. Moreover, the value of the 
fraction of the total ionization which passed through the slit 
in any particular position was not constant, but kept varying, 
so that it was often impossible to get the same value twice at 
the same point. For these reasons the values obtained with 
iron do not possess the same definiteness as in the case of 
the preceding metals. A large number of experiments were 
made with iron in the hope of being able to make the con- 
ditions more definite. ‘T'wo different kinds of iron were used: 
(a) strips of Norway iron rolled from wire, (b) strips of 
transformer iron rolled from sheet. The numbers for those 
experiments on Norway iron which led to an estimate of e/m 
are given in the following table:— 


Norway Iron. Positive Ions. 


| z H V. x e/m. 
550 4850 304 1:40 500 
550 4850 223 1:475 410 
455 4650 200 90 O17 
| 


Average values ... e/m=409, Corrected ... ¢/m=650. 
miH= 23°6, m/H= 14:8, 


| 


‘In considering the above numbers it is important to observe 
that it was noted at the time that in the case of the experi- 
ment which gave the lowest value of e/m the conditions were 
exceptionally steady. If this number (317) were taken 
alone we should have for the corrected values of e/m and 
mH the numbers 


elm 503 and mo] Hl 192. 


ae 


- 


Specific Charge of the lons emitted by Hot Bodies. 553 


The experiments with the transformer iron led to the 
numbers in the following table :— 


Transformer Iron. Positive Ions. 


An | H ¥ x e/m. 
“569 4850 314 1°525 535 
569 | 4850 231 1°70 490 
‘569 | 4850 147 / 2°02 44] 
‘DD) : 4850 195 | 1°50 306 
‘67 4850 122 | 1°85 322 
‘567 | 4850 194 | 1°50 337 
‘DOT 4850 / 232 1:35 327 
"B5D | 4850 159 | 1:50 290 
55D | 4850 / 308 1:00 250 


Average values ... e/m=3872, Corrected ... e/m=592. 
m/H= 26°0, m/H= 164. 


The first three of the above values were given by the same 
specimen of iron. It will be noticed that they were much 
higher than any of the others, and we were unable to obtain 
such high values in any of the subsequent experiments. If 
these are excluded the mean of the rest gives e/m 305, cor- 
rected e/m 486, corrected m/H 19:9. The three high values 
obtained from the first specimen alone give e/m 487, cor- 
rected e/m 775, corrected m/H 12°5. 

In several of the experiments with iron a black soot-like 
deposit was observed on the part adjacent to the hot strip 
after the apparatus had been taken down. This was especially 
noticeable in the experiments with both kinds of iron in 
which high values of e/m had been obtained. 

It was thought at one time that the peculiar values of e/m 
obtained from iron might be due to the magnetic properties 
of the metal affecting the applied magnetic field. This does 
not seem to be possible however, since all the experiments 
were mide above the critical temperature at which the ferro- 
magnetic property disappears. Moreover, it does not seem 
as though the magnetic properties of the metal had anything 
to do with the phenomenon, since the value of e/m for 
nickel, nichrome, and steel (see below), all of which are 
magnetic, was the same as for the other metals investigated 
above. 

In fact it seems pretty certain that in the case of iron we 
have to do with more than one source of positive ionization. 


554 Prof. Richardson and Mr. Hulbirt on the 


One of these might well be the same substance as that which 
gives rise to the emission of the positive ions by the other 
metals investigated, and the high values of e/m obtained 
would be due to the presence of some other substance which 
gave rise to ions of smaller mass. The fact that when the 
highest values of e/m were obtained a sooty deposit was 
observed, would lead one to suspect that this substance was 
carbon, which was dissolved in or chemically combined with 
the iron. On this view, it would be necessary to suppose 
that the dissolved or combined carbon was given off in the 
form of positive ions. This view is supported by the fact 
that the value of m/H corresponding to the cases which gave 
rise to the largest values of e/m is almost identical with the 
atomic weight of carbon. Against this we have to set the 
fact that in the former paper ‘experiments were made on the 
positive ions emitted by carbon itself, and these were found to 
possess the same value of e/m as those emitted by platinum. 
This is not, however, of necessity conclusive. It may be 
necessary for the carbon to be dissolved in a metal before it 
can be given off in the ionized form. In fact there is now a 
good deal of evidence to support the general statement that 
when a substance is expelled by heat from a metal in which 
it has been dissolved, some of it is in the form of positive 
ions. One of the authors* has adduced a considerable 
amount of evidence in favour of the view that the per- 
manent positive ionization produced by hot platinum in 
oxygen and hydrogen is caused by the emission of those 
gases from the metal, in which they have been either dissolved 
or absorbed. 

The discussion of the nature of the substance which gives 
rise to the heavier ions from iron may be conveniently post- 
poned until the various substances under investigation can 
be discussed together. 


Tantalum. 


Tantalum was found to resemble gold in so far as it gave 
small values of e/m when first heated and larger ones after- 
wards. It was not found possible to heat tantalum con- 
tinuously for any length of time as it burnt away rapidly, 
even when the pressure was kept below -001 mm. The 
numbers which were obtained are given in the following 
table :— 


* Phil. Trans. A. ccvii. p. 1 (1906). 


Specific Charge of the lons emitted by Hot Bodies. 555 


Tantalum. Positive Ions. 


| z H We x e/im 
a, As eR ee ee ie 
| 449 4650 200 59 145 

449 4650 400 4G 175 

IL 4850 Ae -p T5 Se) 

551 4850 160 925 115 

551 4850 900 1-225 257 

aol 4850 122 1-55 245 

al 4850 160 1-375 253 


Average value ... e/m=189, Corrected ... e/m=301. 
mH= 51, mH= 32. 


The first two measurements were made on one strip and 
the succeeding five on another. The first measurements are 
probably not so reliable as the others on account of the 
smallness of the deflexions. If we take the last three together 
as representing the final steady value we find e/m (average) 
252, corrected e/m 400, corrected m/H 24:2. hereas for 
the initial values the third and fourth give :—average e/m 125, 
corrected e/m 199, corrected m/H 48-4. Here again the 
ratio of the initial and final values is 2 to 1 as in the case 


of gold. 


Tungsten. 


The tungsten used was in the form of filaments taken 
from a new tungsten lamp. On account of the fact that it 
was impossible to obtain the material in the form of strip 
we should expect, as in the case of osmium, that the curves 
would be very flat and the maxima not sharply defined. 
The case was, however, worse than this, because the curves 
-were very irregular in outline and usually exhibited more 
than one maximum. These were not due to the simultaneous 
emission of more than one kind of ion, as they were present 
just the same in the absence of the magnetic field. The 
most plausible explanation seems to be that the positive ions 
are emitted by some impurity which is liable to be irregularly 
distributed about the filament. A number of attempts were 
made to estimate the value of e/m from the displacement of 
what appeared to be corresponding points in the patterns. 
There is a good deal of uncertainty in this procedure, so that 
it is perhaps not to be wondered at that the results obtained 
do not show a very satisfactory agreement. They are given 
in the following table :— 


556 Prof. Richardson and Mr. Hulbirt on the 


Tungsten. Positive Ions. 


zZ 18e V. 2 e/m. 
567 4850 122 1:8 159 
567 4850 194 "85 108 
567 4850 194 "85 108 
567 4850 307 | "55 72 
567 4850 307 "45 48 
567 4850 122 ‘70 47 
467 4650 200 1:20 541 


Average values ... ¢/m=155, Corrected ... e/m=246. 
m/H= 62°5, m/H= 39°38. 
| 
All that we can reasonably conclude from the above 
numbers is that the positive ions from tungsten are of the 
same order of magnitude as the other metals investigated so 
far as their specific charge is concerned. 


Alloys. 

Experiments were made on brass, nichrome, and _ steel. 
These were all found to give only one kind of ion for which 
the value of e/m was near that for platinum. Although both 
nichrome and steel contain iron, neither of them was found 
to give the high values of e/m which were obtained with 
that metal in the purer state. The numerical values are 
given in the following tables :-— 


Brass. Positive Ions. 


ei. H. V. fy e/m. 


ee — eS 


"583 4850 305 1:05 225 


Values ... e/m=225, Corrected ... e/m=358. 
mlH= 48, miH= 27. 


Steel. Positive Ions. 


Average values ... e/m=216, 
iit) tie 


e. H. Vy. x. e/m. 

567 4850 41 2°50 194 

9 Mh 4850 81 1:85 214 

"567 4850 41 2°75 239 
Corrected ... e/m=343. 


m/H= 28:1, 


Specific Charge of the Ions emitted by Hot Bodies. 557 
Nichrome. Positive Ions. 


| | 
! 


2. Hi: Vi | bi e/nv. 
563 4850 122 | 1:575 233 
563 4250 160 | 1-475 268 
‘563 4850 200 | 1:30 273 | 
‘563 4850 122 1725 280 


| Average values ... e/m=264, Corrected ... ¢/m=420. 
m/H= 36°6, mH= 28. 


Summary. 


In order to see at a glance the results which have been 
obtained, the corrected values of e/m and of m/H are collected 
together in the following table. Where the substance gave 
more than one value of e/m and m/H the abnormal values 
are also shown in the first two columns marked ‘“ initial 
values.” The term initial value seems strictly applicable in 
the case of gold, and probably also tantalum, but in the case 
of iron we are not sure that the high values of e/m were 
really initial values. With this metal, the values obtained 
seemed to depend more on the specimen used than any other 
factor that we could discover. In the case of tungsten the 
values are so erratic that we have made no attempt to dis- 
tinguish between them, although there was some slight 
indication of an approach towards higher values of e/m with 
continued heating. The last two values for platinum and 
the carbon are taken from the previous paper. 


| Initial Initial Permanent | Permanent 

| mubstanee. value of e/m. | value of 7/H.| value of e/m. | value of m/H. 
ePPRTOMUTA doce caie|| |” | na raee ease ges eh 387 25°0 
Pataca .sckescsal nines ert 307 28-7 
COPPOE sconce.) eae iar ee 366 26°4 
gee es re ae PE orl 342 28°3 
LCT GR ay ie tereetincs 380 25-4 
ROSIIIEE oo ocho cscs. eee eee ek ete 420 23°0 
(570) 00h Cees ae 221 43°7 445 21°7 
MOIR? Weed cs saxaa «we 717d 12°5 486 19:9 
Cana LWW. vvlas ss oie 199 48°4 400 24°2 
PB Waaben) 22.0 0~. 246 Sa oe SOAR RAS OL VES) | 
Brass 0212 Pe. cb) OD ay iN eae 358 27:0 
RSME MA ccc menisci eadaiaaeceuetll AN [po 4 nneee ate 3438 28°1 
PUT CWMOVPELA Slik ii ul' | oaueeeade <p Rl ke meee 420 23 0 
Piotimn .cect- + eee eee 384 20°7 


CA Oy ees Gael en pees ih. Se eee oa 27°6 


558 Prof. Richardson and Mr. Hulbirt on the 


With one or two exceptions the numbers in the last column 
do not differ from one another by more than the error of 
observation. There is a considerable error in these measure- 
ments arising from the curvature of the strip when heated. 
Although the strips were very short (‘5 cm. in length) the 
effect of this on z is important since the value of e/m involves 
the fourth power of <. An attempt was made to allow for 
this by direct observations of the displacement of the strips 
when heated, and they were always arranged so that they 
curved in towards the plates. 

The mean value of all the numbers in the last column 
counting platinum only once and equal to 25°35 is 25:3. It 
is probable that the value for iron is affected by the presence 
of some of the substance which gives rise to the ionization 
with very high values of e/m. Jf we omit iron from the 
average we find the mean value of m/H=25:7. This number 
is very close to the values found for carbon and platinum in 
the previous paper, so that the greater part of the discussion 
there applies equally to the present results. The case against 
the view that this ionization is due to one or more of the 
gases whose molecular weights are about 30 (Os, N., and CO) 
is strengthened hy the present experiments. It is probable 
that traces of carbon monoxide are always present when a 
metal is heated in a vacuum in the kind of apparatus used in 
the present experiments, but it is difficult to see why practi- 
eally all the ionization should be due to this gas in the case 
of all the substances inv estigated. It is known that when a 
great many metals are heated the bulk of the gas given off 
is hydrogen, and one of the authors* has shown that when 
this gas escapes from the platinum a considerable amount of 
it is in the form of positive ions; whereas in none of the 
present experiments were any ions detected for which the 
value of e/m approached that of hydrogen. Moreover, the 
metal most likely to emit carbon monoxide in quantity is 
iron, and this is the one metal for which the value of e/m 
deviated most widely from that corresponding to carbon 
monoxide. 

It is perhaps of interest to remark that a great difference 
was observed in the amount of gas given off by the various 
substances used. Platinum and palladium gave off most, 

‘old gave a much smaller quantity, whilst the amount given 
off by. the other materials was inappreciable. 

The mean value of m/H is very near to the atomic weight 


* O. W. Richardson, Phil. Trans, A. cevii. p. 1 (1906). 


Specific Charge of the Ions emitted by Hot Bodies. 559 


of sodium (23:1), and it seems most probable that these 
positive ions which seem common to so many different sub- 
stances are due to sodium or its compounds which are present 
as an impurity. It seems unlikely that they are due to a 
common constituent of the different substances carrying a 
fraction of the ionic charge, since Dr. F. C. Brown* has 
shown that the positive ions emitted by most of the substances 
examined have approximately the value of the kinetic energy 
proper to the temperature of the metal from which they are 
emitted, on the assumption that they carry the normal charge. 
The fact that the value of m/H tends to run a little higher 
than the atomic weight of sodium, may mean that in most 
cases there is a small amount of potassium or its compounds 
present. 

Additional evidence in favour of the view that the 
positive ionization emitted by metals at low pressure is 
due to the presence of alkaline impurities will shortly 
be presented in a paper by one of the authors, dealing 
with the positive ions emitted by the various alkali 
sulphates. 

Whatever the cause of the positive ionization may be, it is 
clear that it is very intimately associated with the metal. 
For a platinum wire which has been boiled in pure nitric 
acid for hours and afterwards had oxygen deposited on it by 
electrolysis in the same medium, when mounted and tested, 
after washing with distilled water, is found to give a large 
initial positive ionization which decays with time. In fact 
it behaves very like a wire which has not been specially 
treated. This shows that the “impurities” must be very 
deep-seated. 


In conclusion we desire to thank Mr. Irving B. Crandall, 
A.B., graduate student in physics, and Mr. Cornelius Bol, 
research assistant, for their help during part of the investi- 
gation. 


Palmer Physical Laboratory, 
Princeton, N. J. 


* Phil. Mag. [6] vol. xviii. p. 649 (1909). 


on 560 J 


LIX. The Eye as an Electrical Organ. By W.M. THornton, 
DSec.. D.Eng., Professor of Electrical Engineering, 


Armstrong College, Newcastle-on-Tyne™. 


Li N the electromagnetic theory of light the amplitude 
of a progressive wave in a partially conducting 
medium is reduced by dissipation of energy as heat. The 
optical media of the eye are typical conducting dielectrics, 
and the following notes are a consideration of (1) the in- 
fluence of absorption on vision, (2) the reticulation of the 
optic nerve on the inner surface of the retina, (3) the least 
electrical current which can be detected by the eye as light, 
from the point of view of the electromagnetic theory. Apart 
from the physiological question of the degree in which 
sensation is proportional to stimulus, and considering only 
the energy of the wave, the reduction of light in intensity 
between entering the cornea and reaching the retina can be 
found by the expressions given by Maxwell f and Heaviside t 
for the decrease of amplitude of waves in a conducting 
dielectric. 
Ina distance 7 they are reduced in the ratio e~*", where 


uy QQ\2> 2 2 
a=—7[ {14+ (=) \ -1| ) 
r pn 


in which X is the wave-length, n the frequency, and »v the 
velocity of the wave in the medium of specific resistance p. 
At high frequencies, such as those of light, this reduces to 


a =2rv/p. 


In the vitreous humour for example, with p=83 x 10° 
C.G.s. units, and with the velocity of the wave, calculated 
from the refractive index, equal to 2°35 x 10!°cm. a second, 
a is found to be 1'¥, and the reduction of amplitude in 
1:43 cm., the thickness of the vitreous humour in the human 
eye, is to ‘088 of that of the incident wave. The energy of 
the wave, which is proportional to the square of the ampli- 
tude, is therefore ‘0077 of the initial value. 

The following table gives the thickness and resistivity of 
the various media, the former for the human eye, the latter 
for ox eyes, and the reduction of amplitude in each part. 


* Communicated by the Author. 

+ Electricity and Magnetism, vol. ii. § 798. 

{ Electrical Papers, vol. ii. p. 422, See also Lodge, Phil. Mag. April 
1899, “On Opacity.” 


The Eye as an Electrical Organ. 561 


The resistances were measured by the Kohlrausch bridge- 
telephone method on freshly killed, though cold, eyes. 

The high conductivity of the humours, greater than that of 
blood, is remarkable. 


| Medium. | Thickness r. p- Mean.}| a. ar. ence 
BROENES byes access ‘115 cm. 455 | 455 | 03 0345) = ‘968 
Aqueoushumour. 36 92-114 | 103 | 1:37 | -495 ‘610 
| Crystalline lens... “39 650-750} 700 | 0:20 | -078 925 
Vitrecus humour.) 1°43 80-87 Sm 6 | 2°43 088 
| { 


The total reduction of amplitude is the product of all the 
figures in the last column and is equal to -048 ; the energy 
is therefore ‘0023 of that of the incident light. 

The reduction of intensity is independent of frequency, 
that is, of colour. ‘The limitation of vision at the blue end 
of the spectrum is not then due to simple absorption of the 
kind considered, but either to selective absorption—probably 
in the cornea—or to photo-chemical inactivity of the visual 
yellow and purple in ultra-violet light. 

The greater part of the incident energy is absorbed before 
reaching the vitreous humour, but the rate of diminution in 
the latter is so great, that in the case of exposure to very 
strong light, injury is lessened. The chief function of the 
vitreous humour, from this point of view, is to act as an 
absorbing screen protecting the retina from possible over- 
exposure. 

Although absorption would appear to reduce the sensitive- 
ness of the eye by its presence, the structure of the retina 
can by it be much more delicate without risk of injury. 

2. Light falling on the retina encounters first the reticulated 
optic nerve. The reason for the fine subdivision of this is 
not fully known. ‘The size of the fibres is, however, such that 
they are peculiarly well suited, as a bolometer, to receive 
the incident energy. 

The distribution of alternating current in the cross-section 
of a cylindrical conductor is not uniform when the frequency 
exceeds a certain value, which depends upon the conductivity 
and diameter. Since the current density is then greater at 
the circumference, this is known as the “skin effect”; it 
reduces the effective current in the conductor. 

The diameter of the nerve fibres of the retina is such 


Phil. Mag. S. 6. Vol. 20. No. 118. Oct. 1910. 2P 


562 Prof. W. M. Thornton on the 


(about 4 to 10 w), that notwithstanding the frequency 
approaching 10 a second, the skin effect does not occur. 
That is, the current density of the electrical current forming 
the light-wave, which being transverse traverses the fibres 
longitudinally, is uniform over their cross-section. In the 
calculation of this the resistivity of the fibres was taken to be, 
as a lower limit, 20 ohms per cm. cube. The value given 
by Waller* is 200 ohms per cm. cube for muscle and nerve; 
the previous value was chosen to cover any possible increase 
of conductivity in non-medullated fibres. If the inner layer 
of nerve fibre were continuous and of the same thickness as 
the fibres, it would not be opaque to electric radiation at the 
frequency of light, unless by selective absorption. The object 
of the reticulation cannot therefore be only to let the light 
through to the rods and cones. 

From observations given later, it can be shown that the 
energy absorbed in the faintest visible light is not sufficient 
to account for arise of temperature in the fibres of more 
than 107° degree C. a second; and since the eye is able to 
follow rapid flicker, it seems improbable that the visual 
stimulus can be in any way thermal in a medium maintained 
at blood-heat. It is more probable, and it is suggested here, 
that the stimulus caused by the electrical currents in the 
light-wave incident on the fibres may be contributory to vision 
by acting as a continuous ‘“‘messenger”’ to the brain, or as 
the vibrator in a coherer system keeping sensitive the contact 
at the synapse between the retina and the rods and cones. 

3. An approximate estimate of the least current which 
can be detected by the eye as light, may be made by con- 
sidering the distance at which the sun would cease to be 
visible. The mean energy reaching the earth’s surface in 
full sunlight was found by the late Prof. 8. P. Langley to 
be 43x 107° erg per cubic centimetre. In a letter from 
him, shortly before his death, he estimates the energy in the 
visible part of the spectrum from the curves of luminosity 
to be 21 per cent. of the whole. The stellar magnitude of 
the sun according to Pickering is —25:5. The mean of 
Wollaston’s, Zollner’s, and Bond’s values is —26°4. Thus 
with the former value the sun has 4 x 10" the intensity of a 
star of the 6th magnitude, the highest visible by the unaided 
eye, with the latter value 91x10". Taking the former as 
probably the more accurate, the sun would cease to be visible 
at 2 x 10° its present distance f. 

The energy reaching the earth would then be 0°25 x 10-” 


* A.D. Waller, ‘Signs of Life.’ 
+ I am indebted to Prof. R. A. Sampson for the astronomical data. 


Eye as an Electrical Organ. 563 


of Langley’s value, that is 1:07 x 10~” erg per cubic centi- 
metre or 0°32 micro-erg per square centimetre per second. 
The ratio of the visible to total energy can scarcely be the 
same at the very low intensities. Retaining it in the absence 
of more reliable data, the energy in the visible light is 
‘067 micro-erg per squarecm. per second. Itis then reduced 
by absorption in the eye in the ratio ‘0023, giving 15 x 10~* 
erg, or with a velocity of 2°3x10'° centimetres a second, 
6°5 x 10-© erg per cubic centimetre of space at the retina. 

The average energy in unit volume of light-wave in non- 
magnetic media is 4777”, where 7 is the root-mean-square value 
of the current per square centimetre. The current corre- 
sponding to the energy in the visible part of the spectrum 
is then 


i = (6°5 x 10-%/4or)}, 


that is 23 micro-amperes per square centimetre at right 
angles to the wave front. 

Since there is no skin effect, this is also the current- 
density in the nerve fibres. The currentin a fibre of ‘0004 cm. 
diameter would be 2°87 x 107 ampere. This, then, appears 
to be about the least electrical current in a nerve fibre which 
can produce the sensation of light. — 

The current, when viewing white clouds in full sunlight, 
with one’s back to the sun, is about 7:0 x 107? ampere in 
the fibres. 

Taking the value of 200 ohms per centimetre cube as the 
resistivity of nerve, the energy absorbed per centimetre 
length corresponding to a current of 2°87x107}* ampere 
is 131x107 watt, or 131x107" erg per centimetre of 
fibre per second. If each nerve fibre conveys a separate 
stimulus, this is what may be called the least specific stimulus 
required in the mental process of vision; the actual length of 
' fibre in the retinal image is a small fraction of a centimetre. 

Rayleigh * has suggested that the least power required for 
hearing is not very different from that of least vision. From 
Rayleigh’s figures for sound, Lodge f finds this power to 
be 6 micro-ergs per second per square cm. Comparing 
this with the energy entering the eye at least visibility, 
obtained above from Langley’s values, viz. 0°32 micro-erg 
per second per square cm., the eye would appear to be able to 
detect about one twentieth of the energy required for the least 
perception of sound. 


* ‘Sound,’ Article 384, footnote. 
+ Jour. Inst. Elec. Engineers, vol. xxvii. p. 931. 


2P 2 


fh op tony 


LX. The Photoelectric Fatigue of Metals. By H. STANLEY 
ALLEN, W.A., D.Sc., Senior Lecturer in Physics at University 
of London, King’s College *. 


HE recorded facts relating to the diminution of the 
photoelectric activity of metal surfaces with time are 
somewhat confusing and contradictory. The difficulties have 
been in part removed by the researches of Hallwachs and 
his fellow workers tf. Hallwachs maintains that the photo- 
electric “ fatigue”’ is not primarily due to illumination, and 
that the size of the vessel in which the plate is kept affects 
to a marked degree the rate at which the fatigue takes 
place. Ignorance of the latter result goes far to explain the 
contradictions amongst the earlier experiments. 

My investigations, which have been in progress for some 
years past, have led me to the same conclusions; and in view 
of the importance of these conclusions in explaining the 
changes involved in fatigue, it seems desirable to put on 
record a short account of my results. 

In the following paragraphs it is shown that in the case of 
zinc, (1) light is not the primary cause of fatigue, (2) the 
fatigue is practically independent of the electric field, (3) the 
fatigue takes place in an atmosphere of hydrogen as in 
ordinary air, (4) the fatigue proceeds more slowly when the 
plate is kept in a small vessel. 

To explain the last result we are forced to the conclusion 
tnat the fatigue must be due to some substance (ozone, Hall- 
wachs; in the case of zinc, ozone, water vapour, Ullman) 
present in small quantity in the atmosphere surrounding the 
plate. The fatiyue must be associated with the condition of 
the gaseous films on the surface of the plate or with the gas 
occluded in the metal. 

The foregoing remarks apply to fatigue in gases at ordinary 
pressures ; in a vacuum other sources of fatigue may possibly 
be present ft, though recent results tend to show that with a ‘ 
perfectly clean metal surface in a very high vacuum there 
would be no fatigue §. 

* Communicated by the Author. 

+ W. Hallwachs, Phys. Zeit. v. p. 489 (1904); Ber. d. math.-phys. 
Klasse d. Kgl. Stchs. Geselisch. d. Wissensch. zu Leipzig, lviii. p. 341 
(1906); Ann. d. Phys. xxiii. p. 459 (1907); AbA. d. naturwissensch. 
Gesellsch. Isis on Dresden, i. p. 65 (1909). H. Beil, Ann. d. Phys. xxxi. 
p. 849 (1910). E. Ulimann, Ann. d. Phys. xxxii. p. 1 (1910). 

{ As, for example, changes in pressure due to absorption of gas by the 
metal (Dember, Phys. Zeit. ix. p. 188, 1908). A change in pressure due 
to gradual absorption of gas by charcoal at the temperature of liquid air 
may have been the cause of the apparent fatigue of zinc in a vacuum 
recorded in my first paper (§ 12). 

§ Millikan and Winchester, Phys. Rey. xxix. p. 85 (1909). 


On the Photoelectric Fatigue of Metals. 565 


Method of experimenting—The apparatus used in the 
present research was identical with that described in my 
earlier papers referred to below. I desire again to express 
my thanks to the Government Grant Committee of the 
Royal Society and to the Council of King’s College for 
defraying the cost of the greater part of this apparatus. 
The mercury-vapour lamp of fused quartz supplied with 
current from a special set of accumulators was used as a 
source of ultra-violet light throughout this investigation. 
Provided sufficient time (from 20 to 30 minutes) is allowed 
for it to assume a steady state, this gives a sufficiently constant 
stream of radiation. 

The photoelectric current between the positively charged 
wire gauze and the metal plate was measured by means of a 
Dolezalek electrometer in connexion with a suitable condenser. 
Readings of the rate of leak were usually taken at intervals 
of two minutes. Most of the results recorded were obtained 
with a zine plate polished with fine emery-paper; both the 
initial activity and the rate of fatigue showed considerable 
variations from day to day, probably in cofisequence of the 
atmospheric conditions, but the results obtained on any 
particular day were usually concordant. In most cases the 
figures given represent the mean of two or three concordant 
determinations. 


(1) Light is not the primary cause of fatigue. 


In my earlier investigations I found that the rate at which 
fatigue takes place is not much affected by the intensity of 

e illumination * or by the character of the source of light 
(mercury-vapour lamp or Nernst lamp 7). Later experiments 
carried out to determine the influence of light on fatigue 
confirm the conclusions of Hallwachs. The photoelectric 
fatigue of zinc proceeds in darkness almost at the same rate 
as when the metal is continuously exposed to light. 

The results of these experiments are embodied in the 
following tables (I., II., and IIL.). The tests in Table I. 
were made with a zinc plate polished with fine emery and 
rouge, tested in the air of the room, readings of the activity 
being taken at intervals of two minutes. The plate was 
exposed continuously to the mercury-vapour lamp except 
where an asterisk appears in the table. In the latter case 
the plate was shielded from ultra-violet light but not from 
the dim light of the room. The activity is expressed as a 


* H.S. Allen, Proc. Roy. Soc. (A) Ixxviii. § 7, p. 489 (1907). 
+ H. 8. Allen, Proc. Roy. Soc. (A) Ixxxii. § 5, p. 164 (1909). 


566 Dr. H. Stanley Allen on the 


percentage of the initial activity, measured immediately 
after polishing. 


TABLE I. 


Zine plate in air of room. Fatigue period, 16 minutes. 


Togs: 9: 100 100 100 100 100 
(1 eee 75 * 17 76 77 
TELE Bee 66 * * 67 64 
iL nee 57 58 * * 58 
Wi 2c... 54 5d 55 * * 
Wake cepa 49 51 51 48 * 
WEE cues. 47 49 47 44 48 
Li 0 46 46 45 43 44 
Le ae 43 45 42 42 43 


The fatigue is not affected by shielding the zine from 
ultra-violet light for a period of about five minutes, no matter 
at what stage in the process the shielding takes place. 

In the tests recorded in Table II. the zinc plate was 
polished with fine emery-paper only. ‘he initial activity is 
given in arbitrary units in the first row of the table, and 
below is given the activity at the end of a fatigue period 
of 16 minutes, expressed as a percentage of the initial 
activity. 


TasLeE II. 


Zinc plate in air of room. Fatigue period, 16 minutes. 


( Arbitrary units ..., 318 | 290 | 288-| 237° | 211 


| Initial activity 
erento ee: | 100 | 100,| 100 |. 1005)) ta6 


Final activity. Percentage ......... 47 | 44 | 48] 44 | 46 


In test a the plate was exposed to ultra-violet light con- 
tinuously. In test 6 it was shielded from ultra-violet light 
for 5 minutes, while in tests c, d, and e it was shielded 
from ultra-violet light for 15 minutes. 

Experiments were also carried out in a closed testing- 
vessel of brass fitted with a quartz window. The zinc plate 
was polished with fine emery, placed in position as quickly 
as possible, and the first reading was taken two minutes after 
polishing. The initial activity was found by extrapolation, 


Photoelectric Fatigue of Metals. 567 


and in the table has been taken as 100. In experiments 
a, d, and e the plate was illuminated continuously by the 
mercury-vapour lamp. In 6 and ¢ it was in complete dark- 
ness for 9 minutes, as indicated by the asterisks in the table. 
During this period the testing vessel was closed with a light- 
tight wooden cover. 


TABLE III. 


Zinc plate in testing vessel. Fatigue period, 16 minutes. 


a b C. d e 
ih eee 100 1¢0 100 100 100 
Be 2s 91 93 92 9 89 
PER 40. ; 85 88 87 83 80 
Uae aoe 79 * * 79 78 
a 78 * * 78 me 
We 8." 75 2 * vue pal 
G7 ae 73 * * {2 70 
1p ea VAN 73 ao 71 69 
IX. 67 13 74 71 68 


It appears from the results in the table that the fatigue 
takes place in complete darkness, though there is evidence of 
a small increase in the rate of fatigue under the influence of 
the ultra-violet light. 

We conclude that light cannot be the primary cause of 
fatigue, though it may play a secondary part in accelerating 
or retarding fatigue. These secondary actions are illustrated 
in my earlier experiments on amalgamated zinc*, or on 
ae zinc at different distances from the mercury-vapour 
lamp T. 

Hise at arrives at similar conclusions, attributing the 
secondary actions of light to the formation of ozone and to 
the heating of the plate. 

I have also examined the action of Roéntgen rays on the 
plate and could detect no decisive change in the rate of 
fatigue, at any rate for an exposure of one or two minutes to 
the rays from a focus-tube at a distance of about 50 cms. 
Hallwachs § records a similar result. 


* H.S. Allen, Proc. Roy. Soc. (A) Ixxviii. § 11, p. 492 (1907)... 
+ H.S. Allen, Proc. Roy. Soc. (A) Ixxxii. § 2, p. 163 (1909). 

¢ Ullmann, Ann. d. Phys. (4) xxxii. § 5, pp. 15-20 (1910). 

§ Hallwachs, Ann. d. Phys. (4) xxiii. p. 467 (1907). 


568. Dr. H. Stanley Allen on the 
2. Fatigue independent of the Electric Field. 


The rate at which fatigue takes place does not depend on — 
the strength of the electric field applied. This is illustrated 
in Table [V., which contains results of experiments made in 
the air of the room and of others made in the brass testing 
vessei. The zinc plate used was polished with fine emery- 


paper. 
TABLE LV. 


Zine plate. Fatigue period, 16 minutes. 


In air of room. In testing vessel. 

110 volts. 480 volts. |; 110 volts... 480 volts. 
ee ce 100 100 100 100 
iL eee 85 ) 89 95 93 
hes se: &0 83 91 | &8 
1h fig 75d 79 | 88 &5 
Werte CL 75 |! 83 83 
Dae he Ft 68 70 81 80 
OB be 65 66 79 78 
MVE ee ccacs 62 60 77 i 
| BEN sh rccins 59 58 76 76 


Again the percentage fatigue was the same whether the 
gauze was charged (to 100 volts) positively (as of course it 
must be when a measurement of the photoelectric current is 
being made) or negatively. This is shown in Table V. 


TABLE V. 


Zinc plate in air of room. Fatigue period, 16 minutes. 


WM Ties fo eticat 100 100 

II. to VII....| Gauze positive Gauze negative. 
ARE ere 63 62 

| ERE Ao 60 60 | 


It was thought that possibly more effect would be produced 
by keeping the zinc plate at a high (positive or negative) 
potential with the rest of the apparatus earthed. The results 
in Table VI. show no effect of this kind when the potential 
of the zinc plate is + 100 volts. 


Photoelectric Fatigue of Metals. 569 
TaBLe VI. 

Zine plate in air of room. Fatigue period, 16 minutes. 
Nite ccmce i tais os 100 100 100 | 100 

| 
| TI. to VII....; Gauze +ve Zine +ve | Gauze +ve Zine —ve 
| | 
VEIT 2).... Mange," 49 | Lae eae! 


pana 2 46 Ae Na ae 


When the zine plate was under examination in the brass 
testing vessel and was charged positively, a small effect was 
sometimes observed on reversing the direction of the field for 
the purpose of taking a reading of the photoelectric current. 
The effect in question corresponded to a small increase in the 
observed current, the increase amounting to‘ about ten per 
cent. of the anticipated value of the current. The increase 
only persisted for a few minutes after the direction of the 
field was reversed. This effect is comparable with that 
observed by Campbell* in the case of the leak from hot 
bodies, the reading taken immediately after reversal being 
greater than the normal. 


3. Fatigue of various Metals in Air and in 
Hydrogen. 


Zinc.—In my earlier papers (loc. cit.) I have shown that 
the fatigue of zine proceeds in such a way that the activity 
for some hours after polishing can be represented as the sum 
of two exponential terms. This would indicate that after a 
prolonged period the activity would approach asymptotically 
a zero value. It has been found, however, that the zinc plate 
retains a small sensibility after several days (in one case after 
26 days); a result which points to the existence of a state in 
which the zine plate would show a small residual activity. 
This would mean, provided the atmospheric conditions . 
remained invariable, the addition of a small constant term to 
the two exponential terms. 

Experiments were made with the zinc plate when the 
testing vessel was filled with hydrogen, prepared by the 
action of pure hydrochloric acid on pure zinc and dried by 


* N. R. Campbell, Phil. Mag. ix. p. 549 (1905). 


570 Dr. H. Stanley Allen on the 


passing over solid caustic potash*. The fatigue in nydrogen 
was found to be very similar to that in air. 

Some results as to the fatigue of other metals are here 
briefly summarized. 

Stlver.—One of the most interesting cases examined was 
a plate of pure silver supplied by Messrs. Johnson, 
Matthey & Co. The plate was polished with rouge paper 
and put in position in the brass testing vessel. After two 
hours’ continuous exposure to the light of the mercury-vapour 
lamp the activity remained unaltered. 

It is remarkable that Ladenburg {, who carried out fatigue 
experiments in a vacuum, mentions silver as one of the metals 
showing marked fatigue, but it must be remembered that his 
surfaces were “ polished once with emery and oil.” 

When the air in the testing vessel was replaced by hydro- 
gen, the same result was obtained as in air; that is, no 
fatigue could be detected after two hours’ exposure to the 
source of light. 

The same plate tested later in the air of the room showed 
fatigue effects, the activity falling to half its initial value in 
two hours. We have here an example of the influence of 
the size of the vessel to be discussed later. 

Aluminium.—This metal was found to behave in much the 
same way as zinc. Its activity can be represented by the 
sum of two exponential terms. When examined in air in 
the closed testing vessel, using the mercury-vapour lamp 
as the source of light, the first term fell to half value in 
6 minutes, the second in 190 minutes. In hydrogen the 
values were not very different, being 7 minutes and 165 
minutes, respectively. 

Copper.—When a copper plate was examined in the 
testing vessel, the fatigue proceeded slowly from the outset ; 
about 3 hours would be required for the activity to fall to 
one-half of the initial value. In hydrogen the fatigue took 
place at about the same rate as in air. In the air of the 
room more rapid fatigue was observed. 


* It should be noted that the metal plate was polished and put in 
position in the testing vessel, the air was displaced by a current of 
hydrogen, and readings of the activity were commenced a few minutes 
after polishing. It is not probable that the gas was entirely free from 
water-vapour, nor is it likely that the air-film on the surface of the plate 
wasimmediately changed. It is to be wished that experiments could be 
carried out in which the plate should be polished after being placed in a 
good vacuum or in an atmosphere of the gas to be employed in the 
investigation. 

+ E. Ladenburg, Ann. d. Physik, xii, p. 558 (1908). 


Photoelectric Fatique of Metals. 5 


4, Fatigue depends on the size of the containing vessel. 


The influence of the size of the containing vessel on the 
rate at which fatigue takes place was verified during the 
present investigation. Examples of this influence have 
already been recorded in the earlier parts of the paper. If 
we compare the results of Tables I. and II. with those of 
Table III., we find that the fatigue is more rapid in the air 
of the room than in the testing vessel. The same difference 
is also shown in Table IV. Similar effects are mentioned in 
§ 3 with regard to silver and copper. 

Inasmuch as the rate of fatigue varied somewhat from day 
to day, probably being dependent on the state of the atmo- 
sphere at the time, it seemed desirable to have a direct 
comparison between the fatigue in the room and that in the 
testing vessel when the air was in the same condition. The 
results of experiments made on the same day are recorded in 


Table VII. 


TaBLe VII. 


Zinc plate. Fatigue period, 16 minutes. 


In air of room. In testing vessel. 
Emery. | Rovce. EMERY. 
100 100 100 
°6 89 90 
7) 79 82 
72 73 79 
67 67 77 
64 64 75 
59 58 72 
57 50 a 
53 52 69 


The fatigue is diminished by putting the plate in the 
smaller receptacle. Incidentally we notice that the fatigue 
proceeds at the same rate when the plate is rubbed with 
emery-paper only as when this is followed by the application 
of rouge paper. 


Conclusion. 


There has been much discussion as to the nature of the 
change associated with photoelectric fatigue. The principal 


972 The Photoelectric Fatigue of Metals. 


views of the character of the change may be summarized as 
follows :— 


1. A chemical change such as oxidation of the surface. 

2. A physical change of the metal itself, as for example 
a roughening of the surface. 

3. An electrical change in the formation of an electrical 
double layer (Lenard *). 

4. A disintegration of the metal due to the expulsion of 
electrons by light (Ramsay and Spencer f). 

5. A change in the surface film of gas or in the gas 
occluded in the metal (Hallwachs). 


Hallwachs has shown from the behaviour of copper and 
its oxides that oxidation cannot be the cause of fatigue, and 
the results of the present paper confirming those of other 
observers are inconsistent with the second, third, and fourth 
views. We must therefore conclude with Hallwachs that 
the main cause of photoelectric fatigue is to be found in the 
condition of the gaseous layer at the surface of the plate. 
This does not exclude the existence of secondary causes of 
fatigue in particular cases. 

The fact that the activity at any instant can be expressed 
by means of exponential terms is not inconsistent with the 
theory put forward. It is sometimes assumed that equations 
of the type here indicated necessarily refer to unimolecular 
changes. But in certain cases reactions are met with which, 
though really polymolecular, behave like unimolecular re- 
actions $. Thus certain gaseous reactions take place on the 
surface of the walls of the containing vessel, and the velocity 
of the reaction is proportional to the pressure of the gas. 
The chemical change then appears as a reaction of the first 
order. Thus a purely surface action may simulate the 
character of a unimolecular reaction. 

In conclusion [ may be allowed a few words of personal 
explanation. I have been represented as a supporter of the 
theory that photoelectric fatigue is due to a kind of radio- 
active change induced by light. Although when my first 
paper was written I was prepared to recognise the possibility 
of this explanation, I was careful to state that the nature of 
the modifications § of the surface suggested was left an open 
question. ‘The view which I favoured for a long time was 
that these modifications of the surface corresponded with the 


* P. Lenard, Ann. d. Phys. viii. p. 196 (1902). 

+ Sir W. Ramsay and J. F. Spencer, Phil. Mag. [6] xii. p. 397 (1906). 
{ H. M. Dawson, Nature, lxxi. p. 582 (1905). 

§ “ Whether physical or chemical modifications of zinc.” 


Electrification due to Heating Aluminium Phosphate. 573 


amorphous and crystalline phases described by Beilby *, 
photoelectric fatigue being a gradual change from the amor- 
phous to the crystalline form through an intermediate (labile) 
phase. 

The experiments described in this paper show that such a 
view is untenable, at least in this simple form, and that we 
must look to the gaseous films on the surface of the metal 
for the explanation of the chief effects of photoelectric 
fatigue. : 

Wheatstone Laboratory, 
University of London, King's College. 
June, 1910. 


LXI. Positive Electrification due to Heating Aluminium 
Phosphate. By A. E. Garrett, B.Sc.t 


[Plate XI] 


I. Introduction and Experimental Arrangements. 


| Se a paper published in the ‘ Philosophical Magazine’ for 

October, 1904, by Dr. R. S. Willows and myself, it was 
found that the halogen compounds of zine when heated are 
able to discharge both positively and negatively electrified 
bodies. A more detailed examination of this phenomenon 
was subsequently carried out by one of us f. 

In those experiments the temperature to which the salts 
were raised was in no case higher than 360° C., and no series 
of observations at pressures lower than a few mm. were 
undertaken. 

Sir J. J. Thomson $ made some experiments to determine 
whether the base or the acid is instrumental in producing the 
ionization, and came to the conclusion that the nature of the 
ionic charge is determined by the acid. Thus he found that 
phosphates when heated produce a very large excess of 
positive ions, halogen compounds produce an excess of 
positive ions, and nitrates an excess of positive at first, but 
when heated sufficiently to be converted into oxides they 
produce an excess of negative. Incidentally he found that 
aluminium phosphate gives off a very large excess of positive 
ions. 

Now the halogen and other compounds used in the previous 

* G.T. Beilby, Phil. Mag. viii. p. 258 (1904); Proc. Roy. Suc. (A) 
xxii. p. 227 (1905); Ixxix. p. 463 (1907). | 

+ Communicated by the Physical Society : read June 10, 1910. 


t Garrett, Phil. Mag., June 1907. 
§ Cambridge Phil. Soc. Proc., p. 105, 1907. 


5TA Mr. A. E. Garrett on Positive Electrification 


experiments are known to be bodies which undergo decompo- 
sition when strongly heated, and the ionization in this case 
may be due to chemical action. Aluminium phosphate, on 
the other hand, is an extremely stable substance, and it is of 
interest to investigate the source of ionization in this instance. 
This salt was therefore chosen for detailed examination. 

The apparatus used is shown in the following diagram. 


(LZ TTT ich EEL 


The glass tube consists of two parts connected by a ground- 
glass joint. The part A is shown in vertical section, the part 
B in horizontal section. e is the strip of thin platinum foil, 
i sq. cm. in area, on which the saltis placed to be heated; the 
temperature of the platinum e is raised bya current, the leads 
for which are the thick copper wires c,d. The electrode E, 
which is connected through a galvanometer to earth, is an 
oblong aluminium disk about 3 sq. cm. in area; the distance 
between the heated salt and this electrode was in most cases 
0-5 cm. f, g are wires of the thermo-electric couple used for 
ascertaining the temperature of the platinum foil with which 
they are fused. To obtain the temperature from the thermo- 
electric current, the deflexion of the galvanometer to which 
leads from the junction pass was noted when a tiny particle 
of K,SO, just melted on the foil. In this way the deflexion 
for two temperatures differing by about 1000° C. was obtained, 
and the temperature in degrees centigrade corresponding to 
any other deflexion could be got by aid of the correction 
curves given by Callendar*. The reliability of this method 
was checked by observing the melting-point of Na,SO,. The 
observations of the behaviour of aluminium phosphate were 
taken over a range of temperature from 900° C. to 1300° C. 

_ Before commencing work with aluminium phosphate it was 
ascertained that only a small current due to ions of either 
sign could be detected when the platinum foil was used 
alone. 


* Phil. Mag. [5] xlviii, pp. 519 e¢ seg. 


due to Heating Aluminium Phosphate. 575 


The sensitivity of the galvanometer was such that a de- 
flexion of one scale division represented a current of 
2 x 10-% ampere. 

When taking readings a saturation voltage, obtained 
from a battery of small accumulators the negative terminal 
of which was earthed, was put on e (fig. 1) as soon as the 
heating of the salt was commenced, and the deflexion of the 
galvanometer noted from time to time. When the voltage 
was taken off e in order to read the thermo-current, time was 
allowed, when the voltage was again put on, for the current 
to become steady before readings were taken. 

When the heating-current had been continued for some 
time, such a large amount of heat had been conducted along 
the copper leads as to cause the melting of the wax which 
was used to render the tube air-tight. This was remedied 
by soldering a small metal tube over each lead as shown in 
fig. 1, and making air-tight wax joints at h and k. 

During the course of the experiments in which the galvano- 
meter was used many attempts under varying conditions were 
made to detect the presence of negative ions, but with no 
success. That such ions are present was afterwards proved, 
but from the results obtained they must be less than 4 per 
cent. of the positive. 

Now if the positive ions are produced by chemical change 
brought about by the heat, then one would expect that a ~ 
decay in the current would take place with the time. It was 
found that the current does decrease when heating is con- 
tinued, so it was decided to obtain the curve of decay for this 
substance. 


Il. Decay of Positive Ionization. 


The aluminium phosphate was made into a paste with 
distilled water and then placed on the foil e; the foil was 
heated slightly by the current before putting it in the tube, 
this causes the phosphate to adhere to the platinum, and also 
gets rid of the excess of water. To lessen the effect of the 
contained gas the tube was evacuated, and the temperature of 
the foil quickly raised to the degree desired. Varying con- 
ditions of temperature and pressure were tested, and it was 
found that the most expedient method was to reduce the 
pressure to about 0°05 cm., and to use a temperature of about 
1200° C. Under such circumstances it was possible to obtain 
a record of the decay in 5 or 6 hours. 

The method of procedure was as follows :—+ 60 volts were 
put on e (fig. 1), and as soon as the temperature became 


i 


576 Mr. A. E. Garrett on Positive Electrification 


steady readings were taken ; the pressure and temperature 
were kept constant throughout. 

Experiments of this kind were carried out with air, 
hydrogen, and CO,, as the gas in the tube. It was found in 
every case when air or hydrogen was used that the manner 
in which the current varied with the time for the first 
20 minutes was peculiar. 

A typical curve to illustrate this is given (Pl. XI. fig. 2). In 
this curve the currents are plotted as ordinates and the times 
as abscissee. 

This shows a rapid fall from A to B, then a rise to a 
maximum at ©, and lastly a decay C to D, in which the 
current decreases roughly exponentially with the time. The 
part of the curve near B sometimes showed still further 
irregularities. 

With CO, as the gas in the tube a typical decay curve is 
represented by ECD. The initial changes observed in air, 
and hydrogen, are apparently due to water, since a pre- 
liminary heating of the phosphate at a lower temperature 
sufficient to expel the water, gets rid of them altogether. 

Further, in the case of CQ,, that gas may possibly assist 
in the removal of hygroscopic moisture, and so prevent its 
action on the salt. 

After some hours’ heating a more or less steady state was 
reached. This state persisted for some months, nor did it 
regain any activity if dry or moist air was admitted, even if 
left for 2 or 3 days. Only on one occasion was a slight 
temporary regain noted, and in this case the interval was 
16 days. 

If after the steady state was reached the salt was moistened 
with distilled water there was a large increase in the current 
which quickly died away, and in about 10 minutes the steady 
state was again reached. The decay of this, AF (Pl. XI. 
fig. 2), being taken in conjunction with a typical decay curve 
ECD, a curve of the forra ABCD is obtained. 

This is further evidence that the initial changes are due to 
water. These changes will not be further considered. 

A typical decay-curve over a longer period omitting these 
changes is shown in Pl. XI. fig. 3. 

The part of the curve near C is exponential ; the portion 
EDC is very similar to the curve given by Rutherford in the 
2nd edition of ‘ Radioactivity,’ p. 342, for the variation in 
activity of the active deposit of Th due to a very short ex- 
posure to the emanation. In the case of Rutherford’s curve 
this is known to be due to the decay of two substances Th A 
and Th B, the former of which does not produce any rays, 


due to Heating Aluminium Phosphate. DIT 


and the equation of the curve is of the form A(e~**—e~), 
The curve of decay of aluminium phosphate can be represented 
by an equation ot the same form up to the point C, beyond 
this the exponential curve lies below the experimental. 
Neglecting this latter point for the moment, if the physical 
analogy as well as the algebraical one holds, we can suppose 
the first effect of the high temperature is to produce from the 
phosphate and the surrounding gas a substance A which does 
not emit ions, that this next produces a substance B which is 
the agent producing the conductivity. As in Rutherford’s 
case the curve alone does not allow one to say to which sub- 
stance the different X’s refer. 

If the analogy holds further and the salt also independently 
produces QC, the latter supplying ions but decaying so slowly 
that its decay can be neglected, the activity due to C is 
represented by a curve having an equation of the form 
B(i—e*), and the whole curve EDCAB should be capable 
of being fitted by an equation 


A(e7Mt— et) + B(1 —e7*), 


This is actually found to be the case. The values of the A’s 
_ depend of course upon the temperature of the salt. 

In PI. XI. fig. 3 the dots represent experimental points, the 
©’s points calculated from the above formula. 

It can be seen from the curve that immediately after the 
very rapid decay there is a somewhat steady state which is in 
turn succeeded by a gradual rise in the current to a final 
steady state. Owing to the small scale used, this is not well 
shown in the curve at AB, although in the experiment there 
illustrated the actual rise amounted to about 20 per cent. of 
the previous steady values. The upper curve AB represents 
this section on a larger scale and makes this point more 
distinct. } 

The relatively large current at the beginning of the heating 
appears to depend very largely upon the nature of the gas in 
the tube, while the final steady current is due entirely to the 
heated salt. 

The results obtained with hydrogen were always of a most 
irregular nature. The rate of decay was also much slower 
in that gas than in air or CO,. It was quicker in CO, than 
in alr. 

No alteration of the yas contained makes any difference to 
the current when in the steady condition if the pressure and 
temperature are kept constant. Further, although an in- 
creased current is obtained with higher temperatures, and an 


Phil. Mag.8. 6. Vol. 20. No. 118. Oct. 1910. 2 Q 


578 Mr. A. E. Garrett on Positive Electrification 


alteration in the pressure produces also an alteration in the 
current, yet on bringing both temperature and pressure to 
their former values, the same steady current is obtained. 


III. Attempts to increase the Activity. 


Richardson * found that a platinum wire which is heated 
to such a temperature that an excess of positive ions are given 
off, gradually becomes less and less active in this respect until 
a more or less steady state is reached. When the wire 
reaches this state he found that it could be rendered much 
more active by passing, for about a minute, an electric dis- 
charge through the tube containing the wire, and he further 
showed that this regained activity persisted for a considerable 
time after the discharge had ceased. 

It was thought that a discharge might havea similar effect 
upon the heated phosphate when it had reached the steady 
state. The heating-current was therefore cut off and a dis- 
charge from a Ruhmkorff coil sent through the tube. This 
increased the current temporarily. If the coil was again 
applied less effect was produced until, after several repeti- 
tions, the coil produced no effect. 

The actual period during which the discharge lasted was 
gradually increased, and it was found that the maximum 
effect was attained when the discharge continued for $ minute. 
On no occasion did it require more than 20 minutes again to 
reach the original steady state after discontinuing the dis- 
charge. In order to ascertain whether the temporary 
increase in current was due to the effect which the discharge 
produced upon the salt itself, or upon the residual gas, a 
fresh supply of CQ, was admitted when the state at which 
the coil discharge produced no effect was reached, and the 
tube pumped down to 0°12 cm. pressure. The coil discharge 
again caused a temporary increase. The direction of the coil 
discharge did not influence the result. 

The extra current therefore appears to be due to an effect 
produced by the action of the discharge upon the surrounding 


gas. 


IV. Action of Heat in the absence of Electrostatic Field. 


When the electrode e was insulated while the heating was 
continued an abnormally large current was obtained so soon 
as the field was put on between eand E (fig. 1). This current 
gradually decayed, but it was some 2 or 3 minutes before it 


* Phil. Mag. [6] pp. 98 et seqg., 19038. 


due to Heating Aluminium Phosphate. 579 


reached its normal value. The magnitude of the increased 
current was found to depend upon the time during which e 
was insulated. It gradually increased with the time, and 
reached a maximum when the insulation had lasted ten 
minutes. Any longer period of insulation was found to have 
no further increasing etfect upon the current. 

It is possible that during the time of insulation positive 
ions are being freed in the salt on ‘e,’ these accumulate in 
the substance and are prevented from escaping by a discon- 
tinuity of the potential at the surface. This continues until, 
at the end of 10 minutes, the field due to the accumulated 
charges is sufficient to take them over as they are formed, 
-when of course no further increase in the number accumu- 
lated takes place. 

When the field is again put on these ions are of course 
dragged out, and the initial value of the increased current 
will depend upon the number of ions which have accumulated. 

This effect greatly increased the labour of taking readings 
under varying conditions, as considerable time had to be 
allowed after a change had been made for the current to 
become steady. These experiments were carried out after 
the steady state had been reached. | 

This effect was more apparent in some gases than in others, 
of those tried it was most marked in the case of CQg. 


V. Effect of Pressure upon Current. 


When the salt is in the steady state it is most convenient 
for investigating the changes due to alterations of pressure 
and temperature. 

The changes in the current when the temperature was kept 
constant while the pressure was varied were first undertaken. 
The contained gases used were air and CO,. The results ob- 
tained were of a similar nature for both these gases. 

Some typical curves are shown in the accompanying 
diagram (PI. XI. fig. 4)in which the pressures are plotted as 
abscissee, and the currents as ordinates. 

From these it can be seen that starting from zero pressure 
there is a very rapid rise in the current in all cases, the 
higher the temperature the more rapid being the rise. For 
each temperature the current reaches a well-marked maximum 
value. This, again, is more pronounced at the higher 
temperatures. When the maximum current has been 
obtained, any further increase in the pressure produces an 
immediate and rapid decrease in the current. For the 
temperatures investigated, this decrease continues until the 


_ od 


580 Mr. A. E. Garrett on Positive Electrification 


pressure attains the value 5 or 6 cms., after this the decrease 
which takes place in the current for any further increase in 
the pressure is always smaller but is still quite marked. The 
rate of decrease is always more rapid at the higher tempera- 
tures. 

When the pressures for which the currents have maximum 
values are plotted against the temperatures, it appears as if 
the pressure and temperature are connected by a straight-line 
law, and that if the temperature could be pushed to a 
sufficiently high degree the largest current would be obtained 
in the highest attainable vacuum. , 

The fact that the maximum currents obtained at the higher 
temperatures and lower pressures were always greater than 
those obtained with the lower temperatures and higher 
pressures is further evidence in support of this view. 

It is quite possible, however, that the slope of the tempera- 
ture-pressure line may gradually become less and less, and 
that it never actually reaches the zero pressure line. 

The peculiar manner in which the current varies with the 
pressure under these conditions cannot be explained by the - 
collision of moving ions, as in all cases the voltage used was 
that corresponding to the flat part of the saturation curve. 
Evidence given later suggests that neutral doublets, such as 
Righi suggested compose the magneto-cathode rays, and Sir 
J.J. Thomson * found indications of in his experiments on 
positive electricity, are driven off when the salt is heated, and 
the current may be due to the break up of these doublets by 
collision with the gas molecules. 

These doublets would be shot off with relatively large 
velocities at the lower pressures, but would have small chances. 
of colliding with gas molecules or other doublets, and so 
relatively few free ions are produced. At higher pressures. 
the doublets would have much slower speeds but greater 
chances of collision. Somewhere between these extremes the 
most favourable conditions for obtaining a maximum current 
may be looked for at each temperature. 

At the higher temperatures the velocity of ejection is. 
greater, and a less number of collisions will be reyuired in 
order to split up the doublet, hence the maximum current is 
obtained under such conditions at a lower pressure. 

With an increase of pressure more collisions are possible, 
but less doublets escape from the heated salt since the 
temperature is lower, hence the current obtained is smaller. 

The great drawback in the way of this explanation is the 


* Phil. Mag. xviii. pp. 828 et seg., Dec. 1909. 


due to Heating Aluminium Phosphate. 581 


very small current due to negative ions, as compared with 
that carried by positive ions, which can be obtained under 
these conditions. | 


VI. Effect of Temperature at Constant Pressure.— 
Phosphate in Steady State. 


Richardson * has proved that the formula l=a@2e-®”9, 
where [=saturation current, and @=the temperature in 
degrees absolute, while Q=a measure of the energy associated 
with the liberation of an ion, represents the connexion between 
the saturation current and the temperature, for positive as 
well as for negative ions given off by heated platinum 
wires. 

The same law holds for various chemical compounds which 
have been tested up to a temperature of 360° C. about f. 

The following resulis have been obtained by heating 
aluminium phosphate in CO, at 0°05 mm. pressure. 
Different quantities of the salt having been used in these 
experiments the absolute values of the currents are not 
comparable. 


y i ‘RE. 
Temperature,| Current, 2x 10-9 Renee Current, 2x 10—9| 
aU. amp. as Unit. | 2 amp. as Unit. | 


— eS _ SS 


| 
| 


880 1 1036 | 1 

950 4 | 1088 3 

970 7 L tIS6 5 | 

995 15 1160 8 

1030 35 | 1195 | 15 | 

1055 49 | 1230 | 34 

1095 103° 1245. | 35 | 

1110 126 1995 | 74 | 
| 1380. | 168 | 


The diagram (PI. XI. fig. 5) shows the results obtained when 
1/0 is plotted against }log.@—log.I. The points are fairly 
evenly distributed about straight lines, and these lines are 


* Roy. Soc. Phil. Trans. A. 207, pp. 22 et seg. 
t+ Garrett, Phil. Mag. June 1907, pp. 732 et seg. 


582 Mr. .A. E. Garrett on Positive Electrification 


parallel to one another. Thus the two sets of readings are in 
accord with one another, and the law may be looked upon as 
true for positive ions given off by heated aluminium phosphate 
in an atmosphere of CO., up to a temperature of 1300° C. 

When the tube was filled with hydrogen gas at a pressure 
of 0°05 mm., and the results obtained were plotted in a 
similar manner, the same law was found to hold for that gas 
up to 1300° C. (see Pl. XI. fig. 5, curve iii.). 

The value of Q may be obtained direct from the diagrams 
by multiplying the pneu of the angle which the line makes 
with the 1/0 axis by 2 

By this means it is found that for the temperatures ranging 
from 900° C. to 1300° C. when the surrounding gas is CO, 
at a pressure of 0°05 mm., the value of Q is 7-1x 104, while 
with the hydrogen gas under similar conditions of pressure 


the value of Q is only 5°3 x 10* for temperatures ranging 
from 1095° C. to 1300° CG. 


VIL. Determination of e/m. 


Sir J. J. Thomson’s cyecloid method* was used for this 
purpose. In this method the ions move in a gas at very low 
pressures under the influence of a magnetic and electrostatic 
field acting at right angles to one another. For a given 
electrostatic field the magnetic field was altered until it 
caused an appreciable diminution of the current passing to 
the electrode E (fig. 1). These experiments could not be 
pushed far because a magnetic field of sufficient strength 
could not be created. The magnet used produced a field 
of 800 gausses. The distance between the electrodes was 
0°45 cm., and the air pressure in the tube was less than 
0-OL mm. When the lower electrode was at a positive 
potential of 6°3 volts the magnetic field produced a decrease 
of about 10 per cent. in the current. 

Thomson has shown that in the case of ions starting from 
a given plane the value of e/m for these ions may be found 
from the formula e/m=2V/H?d?, in which V is the voltage 
to which the electrode from which the ions start is raised, 
H the value of the magnetic field, and “d” the distance 
between the electrodes in cms. 

Substituting the above experimental values in this equation 
we find that e/m= 9700 about. Similar values were obtained 
from other experiments. 

This value of e/m refers, of course, to the lightest positive 
ions present. 


* “Conduction of Electricity through Gases,’ 1st edition, pp. 107 e¢ seq. 


due to Heating Aluminium Phosphate. 583; 


Thomson found for the positive ions from hot platinum 
values ranging from 60 to 720. 

The value of e/m for the hydrogen atom in electrolysis is 
taken as 10*. From this it is seen that these positive ions 
are comparable in size with the hydrogen atom, if we assume 
the same value for “‘e” in both cases. 

A quite appreciable though smaller diminution of the 
current was also obtained under such conditions that e/m 
when calculated was found to be some 3 or 4 times as 
large as given above. Hither the ions affected in this case 
have a mass smaller than that of the hydrogen atom, or else 
their velocity is much less than that due to the electrostatic 
field applied ; in the latter case we might assume that they 
started as free ions at some point between the two electrodes 
and not as such from the surface of the heated salt. 


VIII. Velocity with which ions are shot off from 
the Salt. 


Harlier in the paper it has been suggested that some of 
the ions escape from the salt on account of their kinetic 
energy without the application of an electrostatic field. 

To put this in evidence a Dolezalek electrometer was 
substituted for the galvanometer, since the ions so escaping 
could not be detected with the latter. The sensitivity of the 
electrometer was such that 1 volt produces a deflexion of 
180 scale-divisions. The pressure was reduced to 0:01 mm. 
and the temperature varied as required. 

A weak magnetic field was imposed so as to remove any 
effect which negative ions might produce, and both H and 
the lower heated electrode are earthed initially. 

Ii was then found to receive a positive charge on discon- 
necting it with earth. | 

If the lower electrode was now given a negative potential, 
this had to be raised to 1°2 volts to stop altogether the 
charging up of E. 

If V = the negative potential to which the electrode e is 
raised to prevent positive ions leaving it, e = the charge on 
an ion, m = its mass, and » = its velocity; then, from 
Ve= sm" we can calculate the velocity “v” with which 
these ions are ejected from the heated salt. 

Taking the value of e/m obtained earlier, we find that 
v=1'4x10° cm. per sec., a velocity comparable with that 
(10°) of the positively charged particles which constitute the 
anode rays. fi aed 

With weak magnetic field of too low a value to affect the 


————— =a 


eee 


- 


584 Mr. A. E. Garrett on Positive Electrification 


positive ions, the charging up effect was in every case 
increased, the final deflexion of the electrometer being 
always greater when the field was on. 

When, however, a field of 800, which had previously been 
found to ‘produce a measurable decrease in the current due 
to positive ions, was used, it was found that with tempera- 
tures below 1050° ©. about, the rate of charging up of the 
quadrants was diminished when the field was on. As the 
temperature was reduced below this limit, the effect of the 
field became more marked. ‘This was tested to temperatures 
about 950° ©. 

Above 1050° C. the magnetic field caused an increase in 
the rate at which the electrometer was charged up by the 
positive ions, and when the temperature had reached 1200° C. 
about, the rate with the field on was twice as rapid as when 
the field did not act. 

These increases in the rate of charging up can be explained 
by the fact that negative as well as positive ions are produced 
under these conditions. ‘These ions would cause the rate at 
which E charged up to be smaller than if positive ions alone 
were present, so when they are prevented by the magnetic 
field from reaching the electrode the rate at which it charges 
up increases. This, however, does not occur until the number 
of negative ions which are deflected is in excess of the number 


of deflected positive ions, and this state is apparently not 


reached until the temperature is above 1050° C. 

It must be understood that the actual number of ions 
present under these conditions was very much less than in 
those cases in which the galvanometer was used, since no 
indication of ions of either sign was then obtained in the 
absence of an electrostatic field. 

From 1050° C. to 1200° C. the rate at which E was 
charged up in the absence of the magnetic field was practi- 
cally constant, while the rate with the field on gradually 


‘Increased. 


This would occur if the actual excess of positives which 


‘reached E per second remained constant, and for this to be 
‘the case, since more ions are now present, negative and 
‘positive ions must be formed in equal quantities, such as 
might happen when neutral doublets split up. 


It may be that salts which give off an excess of positive 


‘ions when heated, at first eject positive ions only. Next, it 


would appear that doublets are ejected also, and when the 
salt is raised to a positive potential, or is at zero potential, 


the quickly moving positive ions may by collision with the 


due to Heating Aluminium Phosphate. 585 


doublets cause them to break up, hence causing a large 
positive current to pass between the electrodes. The initial 
positive ions appear to increase in the number given off 
per second as the temperature is raised until about 1050° C., 
after which the output appears to remain constant. 

If, on the other hand, the salt is raised to a negative 
potential, the initial positive ions are prevented from leaving 
it, and so at reduced pressures, with electrodes a very small 
distance apart, the doublets have smaller chances of breaking 
up; thus the current when the salt is negatively charged 
is relatively small compared with that due to positive: 
ions. 

That the potential to which the salt is raised has an 
important bearing upon the relative number of positive and 
negative ions present is clearly shown by the fact that the 
current due to negative ions is not so small compared with 
that due to positive ions when the salt is at zero potential 
and the gas pressure is low. 

Also, at atmospheric pressure when the products due to 
heating the salt were removed by an air blast, there was 
always quite a large current produced by the negative ions. 


IX. Nature of the Ions at Atmospheric Pressure. 


For this purpose the apparatus used was two brass tubes 
of 1:7 cm. internal diameter, insulated from each other, and 
each having an insulated wire 0°33 cm. diameter and 28 em. 
in length along the axis. These wires could be in turn con- 
nected with a Dolezalek electrometer of such sensitivity that 
one volt caused a deflexion of 820 scale-divisions. 

The method adopted was to heat the phosphate on platinum 
foil through which a current was passed, and to suck air 
past the heated salt, and then through the two brass tubes 
placed one behind the other. 

In order to avoid great fluctuations in the temperature, 
the platinum and its leads were enclosed in a wide glass tube, 
one end of which was connected air-tight to the testing 
apparatus, and the other closed loosely with cotton-wool. 

Readings were taken after the salt had been heated for a 
sufficiently long time to bring it to the steady condition. 

The electrometer showed no initial leak even when one of 
the electrodes was connected with its quadrants, and the tube 
containing the electrode was raised to a positive potential of 
650 volts. A saturation current was obtained when the 
electrode nearer the heated salt was connected with the 


586 Mr. A. E. Garrett on Positive Electrification 


electrometer and a potential difference of 314 volts was used. 


This is shown by the following Table :— 


Current due to Current due to 
Volts. positive ions in Volts. positive ions in 
Arbitrary Units. Arbitrary Units. 
42 46 390 130 
84 58 432 130 
126 79 474 128 
168 85 516 130 
210 95 558 126 
236 111 608 130 
275 123 656 130 
314 130 
348 130 


Under the conditions of experiment (velocity of air through 
the tube being 40 cms. per sec.) it can easily be calculated 
from the formula 

_ (b?—a?) log.b/a 
Ui) 2VE 
in which 


v=velocity of ions in cms. per sec., 
b=radius of the tube, 
a=radius of wire electrode, 
V =potential-difference in volts between wire and tube, 
t=time taken by air to pass from one end of the elec- 
trode to the other, 


that all ions with velocity greater than 0°0027 cm. per sec. 
are withdrawn when the saturation voltage of 314 volts is 
put on. When the voltage is raised to 656, ions must have 
a velocity less than 0:001 cm. per sec. to be able to escape 
from the first tube. 
If the front electrode (i.e. one nearer the heated salt) is 
earthed and the front tube kept at any positive potential 
reater than 320 volts, no current should be found near the 
back electrode when this is connected with the electrometer, 
and the back tube raised to any positive potential ; a current 
could, however, be detected. tg 
Even when both tubes were raised to a positive potential 


due to Heating Aluminium Phosphate. 587 


of 656 volts there was a current of 8 or 9 arbitrary units. 
The total current on the back electrode when the front tube 
was earthed was found to be 118 units. Thus about 8 per cent. 
ot the total current here appears to be due to ions which are 
too slow moving to be extracted even with the very high 
voltage used, or which have been formed after passing the 
front electrode. 

It may be mentioned that it was necessary to raise the 
back tube to a high potential before any indication of these 
extra ions was forthcoming. Thus with 432 volts on the 
front tube a current could be just detected when 190 volts 
was put on the back tube. 

If the current of 8 units on the back electrode is due to 
ions which are too slow to be stopped by the field on the 
front electrode, then some indication of these would be 
expected on the saturation current curve, 7. e. after passing 
314 volts the curve should still gradually ascend. 

Nothing of the kind, however, takes place, and, as is seen 
from the table of observations given, a small increase could 
easily have been detected. 

If we assume that neutral doublets are present which 
break up into ions after passing the first electrode, then the 
results are readily explained. 

Should the extra ions be in reality due to the splitting up 
of neutral doublets rather than to the presence of ions of 
extremely low velocity, one would expect to find an equal 
number of negative and positive ions formed after passing 
the first electrode, and this should occur although the satu- 
ration currents on the front electrode due to positive and 
negative ions may differ widely. For aluminium phosphate 
these saturation currents on the front electrode are in fact 
very different. It was found that a negative voltage of 220 
on the front tube was sufficient to obtain 2 saturation current 
with negative ions. When both tubes were brought to a 
negative potential of 656 volts, the front electrode being 
earthed, and the back electrode connected with the electro- 
meter, there was a current of 8 units—z.e., precisely the 
same as that obtained with positive ions under exactly similar 
conditions. 

This appears strong evidence in favour of the view that 
the extra ions are due to the splitting up of neutral doublets 
and not to ions of very low velocity. 

When a current voltage curve for the positive ions is 
plotted using the values given above, it is seen that the 
curve formed by joining the points obtained is not of the 
usual type. 


088 Mr, A. E. Garrett on Positive Electrification 


At about 200 volts there are indications that the current 
is nearly saturated, yet on slightly increasing the voltage, it 
again rises very rapidly and does not actually reach its 
saturation value until over 300 volts are put on. 

On the other hand, the curve obtained with negative ions 
is quite a smooth one and does not show any indication of 
saturation at a voltage lower than 220, nor does any further 
increase in current take place if the voltage is raised from 
220 to 656. This is shown in fig. 6, in which the upper 
curve represents the current due to positive ions, the lower 
curve that due to negative ions at different voltages. 


eae 
coon 


CurRenr Are/rTRARy UNITS 


This, taken in conjunction with the fact that a current is 
obtained at the second electrode when 656 volts aré put on 
the first, suggests that either two classes of positive ions are 
present, or else neutral doublets. The former is rendered 
unlikely on account of the constancy of the current which 
passes between the electrodes after 320 volts is reached. 


X. Further Evidence of Doublets. 


In order to test whether neutral pairs as well as ions are 
shot off from the heated salt when surrounded by gas at 
a low pressure, the apparatus shown in the next diagram 
was used. 

The essential part of this apparatus consists of a long glass 
tube, in one end of which is a Faraday cylinder F, and in 
the other end the heated salt on the platinum foil 8. 

During the experiment the foil was kept at a positive 
potential sufficiently high to prevent negative ions from 
leaving the salt. The wire gauze P was also kept at such a 
positive potential as to drive back any positive ions which 
have passed through the earthed metal tube E. 

The apparatus was evacuated to an air-pressure of about 


due to Heating Aluminium Phosphate. 589 


0:01 mm. The outer Faraday cylinder was earthed and the 
inner one connected to the leaf of an electroscope which 
was charged to a definite potential, positive or negative, as 
required. 


Fig. 7. 


To Baitery 


The insulation was such that the rate of leak of electricity 
from the leaf when the salt was not heated was imperceptibly 
small when the charge was of either sign. 

As soon as heating was commenced quite a distinct leak 
was noticed, and the rate of leak was the same whether the 
leaf was charged positively or negatively. 

On the other hand, after the inner cylinder had been 
reduced to zero potential, no charging up could be detected, 
thus showing that there was no excess of free ions of either 
sign in the neighbourhood of the inner cylinder. 

No positive ions could pass through the field between P 
and H, and it is highly improbable that any free negative 
ions leave the salt when the latter is charged positively, as 
in the experiment. Again, the field between P and the outer 
cylinder is a further preventive against any stray negative 
ions passing through the small hole into the space between 
the cylinders. Hence it seems that any ionization produced 
in the Faraday cylinder when the salt is heated, can only be 
brought about by the split up of doublets which have passed 
through the various electrostatic fields and diffused into the 
space between the two cylinders. 

An effect of a similar nature has been noted by Sir J. J. 
Thomson * when working with a hot lime cathode. 


* Phil. Mag. Dec. 1909, pp. 829 e¢ seq. 


990 Electrification due to Heating Aluminium Phosphate. 
XI. Reectifying Eject of the Heated Salt. 


Owing to the great velocity with which the positive ions 
are shot off from the heated salt when the pressure is very 
low, it was thought that such a tube as shown in fig. 1 could 
be used for rectifying alternating currents providing the 
temperature of the salt was kept within certain limits. It 
has been found by various experiments that it can be so 
used. 

Further work on this point is in progress. 


XII. Summary. 


G.) The decay of the current due to positive ions obtained 
by heating aluminium phosphate has been investigated, and 
it is found that the curve connecting current and time can 
be represented by a formula of the type 


A(e7At eA?) 4. B(L —e*2"). 


(ii.) During the first part of the decay, the nature of the 
surrounding gas and the water contained by the salt have an 
important influence. When the steady state is reached the 
gas has no apparent influence, but water still temporarily 
increases the activity. 

(iii.) The discharge produced by an induction-coil tempo- 
rarily increases the current which is carried by the positive 
ions. 

(iv.) When the temperature is kept constant, it is found 
that for each temperature there is a definite pressure at 
which the current is a maximum. This pressure being 
lower, the higher the temperature. 

(v.) The Richardson formula I=a6#e— 9 can be used to 
express the relationship between the current and absolute 
temperature when the pressure is kept constant. 

(vi.) A value is obtained for e/m which indicates that the 
smallest positive ions present at the lowest pressures must 
be of a magnitude comparable with that of the hydrogen 
atom. 

(vii.) The high velocity of the ions at low pressures, aud 
also the fact that some escape with great velocity even when 
no external field is applied, leads one to expect that a tube 
in which some aluminium phosphate is heated might be of 
use as a rectifier for alternating currents. It can be so used. 

(vili.) The experiments (a) with varying pressures at con- 
stant temperature, (b) at atmospheric pressure in which a 
current is produced after all the ions are apparently removed, 


Convection of Heat froma Body cooled by Stream of Fluid. 591 


and (c) in which the charge on a Faraday cylinder leaks 
away when care is taken to prevent free ions reaching it, 
seem to indicate that neutral doublets as well as free ions are 
ejected from the salt. 


In conclusion, I should like to thank Dr. R. 8. Willows, 
in whose laboratory these experiments have been carried out, 
for the interest he has taken throughout the course of this 
research, and Mr. F. C. G. Bratt for help in the construction 
of the apparatus used. 

Cass Technical Institute, 


Jewry Street, E.C. 
May 1910. 


LXIL. The Convection of Heat from a Body cooled by a . 
Stream of Fluid. By ALEXANDER Russe, W.A., D.Sce., 
MIEE., Principal of Faraday House *. 


TABLE OF CONTENTS. 
. Introduction. . 
. Historical. 
. The assumptions made. 
Flow in two dimensions. 
Circular cylinder. 
Cylinder with elliptic section. 
. Flat strip. 
. Cylindrical tube. : 
. Tables of the values of the function Z/X. 
. Simplified formula for cylindrical tube. 
. Turbulent flow. 
. Electric current required to fuse a wire. 
13. Schwartz’s experimental results. 
. Steady temperature of a wire carrying an electric current. 
15. The effect on the convection of heat from a cylinder of 
putting a covering round it. 


4 
© © 00 SIO OVI C9 DO 


bd bt 
bh) 


he 
rm 


1. Introduction. 


A ewe phenomenon of the convection of heat at the surface 
of a body immersed in a cooling fluid is one which does 
not lend itself readily to mathematical calculation. If the 
fluid be a gas the variations of the pressure, density, and 
velocity at different points of the gas so complicate the 
problem that little progress towards a complete solution has 
yet been made. In the case of liquids flowing past a body 
with appreciable but not excessive velocity, Boussinesq f has 
* Communicated by the Physical Society: read July 8, 1910. 


+ Théorie Analytique de la Chaleur, t. 11. 1908, and Journal de Mathé- 
matiques, 6° Série, t. 1. (1905). 


| 
1 


592 Dr. A. Russell on the Convection of Heat 


found some approximate solutions which deserve to be more 
widely known. The author has therefore thought that it 
would be useful to give the proofs in full of the more practical 
of Boussinesq’s formule, laying stress on their limitations, 
and pointing out some of ‘their applications. The author also 
discusses the important problem of the heating, due to stream- 
line convection, of a liquid flowing through a cylindrical 
tube, and gives a table by means of which approximate 
solutions can be found without much difficulty. 


2. Historical. 


The differential equations the solutions of which would give 
the flow of heat through a fluid were first given by Fourier *. 
The fundamental equation was put into a more manageable 
form by Poisson f, but neither he nor Fourier gave any 
solution of it. 

Poisson writes the equation as follows :— 


De = a(S) + By(49y) tet Sey = 


where @ is the temperature of the fluid at the point (a, y, 2), 
c the capacity for heat per unit volume, & the conductivity, 
and D@/Dé the rate at which the temperature of a particle of 
fluid passing through the point (a, y, 2) is increasing in the 
direction of the motion of the fluid at the point. When 
written in this form it is interesting to notice how similar this 
equation is to the equation of the flow of heat through a solid 
body. We may also write 
Dé _ 0a 6 0 00 : 

De oat ud 4 0 ee 540% Alene 
where u, v, and w are the component velocities of the 
current at the point (#, y, z) parallel to the three axes 
respectively. 

In addition to equation (1) we have the ordinary hydro- 
dynamical equations t, namely, the equation of continuity 
and the three equations of Huler. 

A. Oberbeck § discusses the general equations, and gives a 
solution for a special case. In a valuable paper || L. Lorenz 


* Mémoires de l' Académie, t. xii. p. 507 (1820), or Giuvres de Fourier 
(Darboux’s Edition), t. 11. p. 275 

+ Théorie Mathématique de la Chaleur, chapter iv. (1835). 

{ Lamb’s ‘ Hydrodynamics,’ chap. i. 

§ Ann. der Physik, vii. p. 271 (1879). 

| Ann. der Physik, xiii. p. 582 (1881). 


from a Body cooled by a Stream of Flwud. af 


obtains an approximate solution for the case of a heated strip 
cooling in air. When the strip is protected from draughts 
he proves that the heat convected from it varies as 6°/* where 
0 is the difference of temperature between the strip and the 
air before it is heated by the strip. L. Graetz ~* finds the 
mathematical equation for thermal conduction in a liquid 
flowing through a cylindrical tube, and obtains a solution 
in terms of Bessel’s functions. Harold Wilsonf gives a 
solution of this problem, taking the viscosity of the liquid 
into account ; but unfortunately his solution is only applicable 
to a very special case. 

Boussinesq was the first to state clearly the laws for the 
cooling of a heated body by a stream of liquid when the flow 
is not turbulent. In 1901 +t he published the formula for 
the cooling of a strip by a liquid flowing past it in a direction 
at right angles to its length and parallel to its breadth. Four 
years afterwards (I. c. ante) the same author published the 
solution of the problem of the convection of heat from a 
horizontal cylindrical rod of elliptical cross-section immersed 
in a liquid flowing in a direction at right angles to the axis 
of the rod. He also gave the solution for the similar problem 
of the convection of heat from an ellipsoidal shaped body. 


3. The Assumptions made. 


In order to simplify the mathematical work the following 
assumptions are made. The liquid is supposed to be 
athermanous, that is, opaque to heat rays. It is also sup- 
posed to have no viscosity. The liquid therefore slips past 
the surface of the solid. In addition it is supposed to be 
incompressible. Hence we should only expect the solutions 
to give roughly approximate values when applied to the 
problem of spheres and cylinders being cooled by currents of 
air. It is instructive to notice, however, that Boussinesq’s 
result, that the convection of heat by a stream of liquid from 
a sphere or a cylinder maintained at a constant temperature 
varies as the difference of temperature between the solid and 
the liquid and as the square root of the velocity of the current, 
is in good agreement with the results obtained by P. Com- 
pan § from experiments with spheres in draughts of air, and 
also with Kennelly’s || results for the cooling of cylindrical 
wires. Boussinesq’s theoretical results also would lead us to 

* Ann. der Phystk, xviii. p. 79 (1883). 
+ Camb. Phil. Soc. Proceedings, xii. p. 406 (1904). 
t Comptes Rendus, cxxxiii. p. 257. 


§ Ann. de Chim. et Phys. xxvi. p. 488 (1902). 
|| Amer. Inst. Elect. Engin. Proc. July 1909. 


Phil. Mag. Ser. 6. Vol. 20. No. 148. Oct. 1910. 22 


594 Dr. A. Russell on the Convection of Heat 


expect that the loss of heat per square centimetre of the 
surface of a wire would be greater the smaller the diameter 
ot the wire. This is in agreement with the experiments of 
Cardani*, Ayrton and Kilgour f, Sala ¢, and Kennelly §. 

We also show later on that in certain cases the fusing- 
current of wires when immersed in a stream of liquid varies 
as the 125th power of the radius of the wire. This agrees 
with experimental results obtained by Schwartz and James | 
for wires in air. 

The further assumptions are made that the thermal con- 
ductivity of the liquid is very small and that the variation in 
its density does not appreciably alter the shape of the 
trajectories of the liquid particles in the immediate neigh- 
bourhood of the solid from the shape they have during 
isothermal flow. ‘The former assumption is true in most 
practical cases, and the latter is permissible when the 
velocity of the current is appreciable and no eddies are 
formed. 

It is interesting to remember that in Hele-Shaw’s{] method 
of reproducing the stream-lines of a perfect fluid flowing past 
an obstacle in two dimensions, a thin film of viscous liquid, 
glycerine for example, is employed, and results of high 
accuracy are obtained. Even for a thick film, the shape of 
the lines does not alter much from the ideal case, and hence 
the assumption that the stream-lines coincide with the stream- 
lines of a perfect fluid is not a serious one. 

The surface of the solid being cooled by the current is 
supposed to be isothermal, and the liquid in immediate 
contact with it at any instant is supposed to have the same 
temperature as the solid. These two assumptions are quite 
legitimate. 


4. Flow in Two Dimensions. 


Making the above assumptions we shall now cbtain the 
differential equation which determines the temperature at 
any point of the liquid, once the steady state has been 
established. pet 

Let us suppose that the velocity of the liquid at a great 
distance from the solid being cooled is V. In our problem 


* Nuov. Cim. xxx. p. 83 (1891). 

+ Phil. Trans. clxxxiii. part i. p. 871 (1892). 

{ Nuov. Cim. iv. p. 81 (1896). ay 

| Journ. Inst. Elect. Engin. xxxv. p. 364 (1905). 

4] Brit. Assoc. Report, 1898. In this report Sir G. G. Stokes gives a 
theoretical proof of the method. . 9 A115 E OG el 


L.c. ante. 


from a Body cooled by a Stream of fluid. 599 


it is convenient to denote the hydrodynamical stream- 
function by Va, and the velocity-potential by V8. In fig. 1, 


AA’ and BB’ denote adjacent stream-lines, and A'B! and 
AB denote adjacent equipotential curves. Let 


AA! =: Ossy 
and let AB = ost 
Then, by hydrodynamics, 
OV vou ; 
ay ae ee (3) 
where g is the velocity of the flow of the liquid at the point 


(sy, Sq). 

Consider the flow of heat in the time Dé into a prism 
having unit length, and having ABB’A’ for its cross-section 
(fig. 1). We suppose that this prism is moving with the 
liquid, and its velocity is therefore g. Let @ be the initial 
temperature of the liquid inside this prism, and let + D@ be 
the temperature at the time Dt. The gain by the flow of heat 


_ from AA‘ to BB’ during this interval is 


8 (7,99 

35, (45) O51 O52 Di, 
and the gain by the flow from AB to A’B’ is 

0 /,00 

AG cs) OS; OS2 Dt, 


where & is the conductivity of the liquid. Hence, if c denote 

the capacity for heat of the liquid per unit volume, the gain 

of heat by the element contained by the prism is ¢Qs, 0s, D#. 
2. ee 


396 Dr. A. Russell on the Convection of Heat 


We have, therefore, 


COS, OS. DO = {2 Gar + (FS) } Os Os, Dt, 
and thus 


= =2(055)+ a =) ty hae 


Since we suppose that the liquid is flowing in the direction 
AA’, we have, when the steady state is reached, 


ve 09 O82 00 


4 —~— ¢€ ——. 
"De "Os." ot = oh 
and hence, assuming / ee we get 


Oem O70 O78 
Tian aA gat toate een 


As qg varies with both s, a4 So, it appears at first sight as 
if it would be very difficult to obtain a solution of this 
equation. If, however, we alter the variables from s, and s, 
to « and §, the equation simplifies in a remarkable way. 


We have 
30 9032 3028 


95 0205, | OR OS 


and thus 
08. 00 /o¢e vioe 5 One Ce SOR oS 
352 da\Oe,) | “SadBOn On | OAKS 
ov 0 @ 00 os 
0 Os? i ye Or 


. e . 0°70 
A similar equation holds for 6 
Sq 
Noticing that 
04 obadd 0810. Bem. 


05; ai 059 EN? Ose fy, Ost 
\7a = 0} and "\/76 0; 
O78 Ores 026 = 370 
0s, Fee C3 a z+ + 3B) 
We also have 


we get 


06 _ 06 g. 
Oss Lacy 


from a Body cooled by a Stream of Fluid. 597 


Hence, substituting these values in (5), we get 
O00 1. eran 0 0 ) 
age ay 52 + 5B Re uceat baat” #0} 
which is a much simpler equation than (5) as the coefficient 
of the right-hand side is a constant quantity. 
Our assumptions allow us to simplify this equation still 
further. Since the liquid is a very bad conductor of heat, 
@ alters very rapidly with « but very slowly with ~. The 
term 0°0/07 is also negligibly small compared with 9070/0’. 
We thus obtain the equation 
ee yee ee a) 


the solution of which has been put into various forms by 
Fourier and others. 


d. Circular Cylinder. 
We shall now consider the problem of the cooling of a 
circular cylinder immersed in a stream of liquid with its axis 
horizontal and at right angles to the direction of flow. 


Fig. 2 


2, 3, 


fay A, 
Stream lines of Fluid flowing past cylinder: 
[> Ae are the lwo Singular equipolential curves. 


Let us take the origin of co-ordinates on the axis of the 
cylinder, and jet us suppose that the liquid is flowing with 


598 Dr. A. Russell on the Convection of Heat 


velocity V in the direction XO, and that its temperature is 
zero before it meets the cylinder. 
In this case we know * that 


8 
and AH PP ape (8) 
B aS i o- ee 


where a is the radius of the cylinder and 
9 ip? Gay". 


The equation to the stream-line 2), which flows on the 
surface of the cylinder (fig. 2), is # = 0, and the equations 
to the two equipotential lines, 8) and (,, are 


2 2 
(1 ++ 5) =—2q and (4 + =) = 2a 


respectively. These curves cut the cylinder and the stream- 
line # at angles of 45°. We see therefore that the velocity 
of the liquid at L and L’ must be zero. The velocity at M 
and M'is 2V. At a great distance away from the cylinder 
B, and B, practically coincide with the lines 


ea@——2a and «= 2a. 


Let us suppose that the temperature flow has become 
steady and that the temperature of all points on the stream- 
line a is f(8). On this stream-line (fig. 2), from @ = @ to 
B = B,, we have 


6 = f(8) =0, 
and on the same stream-line from f, to 8) we have 
0 = f (8). 


It is easy to verify f by differentiation that 


o=a/2("7(8+ Ee) e 228 aa 


is a solution of (7). Also, when «is zero, 0 = /(f). ‘this 
solution, therefore, is applicable to our problem. 


* Lamb's ‘ Hydrodynamics,’ Third Edition, p. 74. 
+ Cf. Boussinesq, Application des Potentials, p. 560 (1885). - 


from a Body cooled by a Stream of Fluid. 099 

- The flux of heat emitted per unit length of the cylinder 

per second is y—1(2°) Oso, where a is the thermal 
051/9 051/o 

gradient at the surface of the cylinder where «= 0. By 


98) 3. 


means of (3) this may be written in the form }— ks). 


But from (9) we have 


& igi 2/S("/6 + 7°) 0; 


where we have written 7? for cVa*/(2k&?). 
Hence the total flux H of heat per second from unit length 
of the cylinder is given by . 


BW 31 
H=—4 A ae hod '(B + 7°) 0B on 


a1 JN G+ 0) B+ aH (00) 


Now /(8,+7’) is zero from 7 almost equal to nothing up 
to n equal to infinity; and f(8)+77) is % from 9 equal to 
zero up to 7 equal to 8,— >, and practically vanishes for 
all greater values of 7. Hence 


ckV (Bi —Bo 
H= any | f (Bo + 7°) on 
= tn / OX /B BO 
A ee A 


where s is the specific heat, and a is the density of the 
liquid. 

This result, which is true for two-dimensional flow round 
a solid of any shape immersed in a stream of liquid, agrees 
with that given by Boussinesq. It shows that the loss of 
heat from the solid is proportional to the difference of 
temperature between the solid and the liquid. Newton’s 
law is thus verified when the cooling fluid is a liquid. It 
will be remembered that Newton enunciated his law with 
reference to the convection and not the radiation of heat. 
He considered the case of a block of iron being cooled in a 


600 . Dr. A. Russell on the Convection of Heat 


current of air flowing uniformly. He states * “ aeris partes 
aequales aequalibus temporibus calefactae sunt & calorem con- 
ceperunt calori ferri proportionalem.” 

A. C. Mitchell + has shown that Newton’s law is very 
approximately true up to a difference of temperature between 
the solid and the air of 200° C., and P. Compan } has proved 
it true for temperatures up to 300° C. 

In several practical applications the assumption of Newton’s 
law for the convection of heat by fluids leads to results which 
are found to be in close accordance with experiment. For 
instance, in the theory of the Irwin § hot-wire oscillograph, 
the assumption is made that the convection of heat from the 
heated metal strips which are immersed in convection- 
currents of oil is proportional to the difference of tem- 
perature between the metal and the oil. The very satis- 
factory results obtained in practice prove that the assumption 
is approximately correct. It will be seen from § 11 below 
that, even in the case of the turbulent motion of water 
through a pipe, Newton’s law is very approximately true. 
It is not applicable, however, to natural free convection 
from a heated body in a gas or a liquid. In this case || 
Lorenz’s law (see above) is applicable. 

In the case of the circular cylinder we find, from the 
values of 8, and 8p given above, that 


B; — Bo = 4a, 
and hence he 54/e ape ber. 2 (12) 


If we denote the surface of unit length of the cylinder 
by 8, so that S = 27a, the expression for the rate 
at which heat is lost by the cylinder per unit length is 
generally assumed by engineers to be equal to ASO, where h 
is independent of the radius of the cylinder. We see from 
(12) that the value of A for the perfect liquid is given by 


el sak V 
h — a4 ey . . . ° . e (13) 


Thus h varies inversely as the square root of the radius of 


* “Scala Graduum Caloris,” Phil. Trans. p. 828, April 1701. The 
paper 1s not signed. In Newtont Opera Horslett, vol. iv. p. 403 (1782), 
the title is given as “Tabula Quantitatum et Graduum Caloris.” 

t Roy. Soc. Edin. Trans. xl. 1, p. 89 (1899). t L.c. ante. 

§ Journ. Inst. of Elect. Engin. vol. xxxix. p. 617 (1907). Ox. 
fsa Compan, /. c. ante; H. Ebeling, Ann. der Physik, xxvii. 2, p. 891 


from a Body cooled by a Stream of Fluid. 601 


the cylinder, and therefore the assumption that it is constant 
is not permissible. For example, if the radius of one wire is 
a hundred times that of another, the average heat per square 
centimetre of surface which is carried off per second by the 
liquid from the small wire is ten times greater than from the 
Jarge wire. We also see that if we quadruple the velocity 
of the flow of the liquid, the temperature of the wire being 
maintained constant, the convection of heat is doubled, and 
if the convection of heat is constant the difference of tem- 
perature between the wire and the liquid is halved. 


6. Cylinder with Elliptic Section. 


Let the direction of the current be at right angles to the 
axis of the cylinder, and let it make an angle a with the 
major axis of the elliptic section which we take as the axis 


of X. If 


l=cos a and m=sin a, 


we easily find from the formule given in Lamb’s ‘ Hydro- 
dynamics’ (p. 70, 3rd ed.) that 


where 
a=csin£coshyn, y=ccosésinhyn, and c= Va?—6b?. 
On the surface of the cylinder we have 
a=ccoshyn, b=csinh y, 
and thus at these points 
ex=asing, and y=dcosé. 
At points, therefore, on the surface of the cylinder, 


Bal" at latmey+my. 
= (12 +m: 1) (a+b), 
a l 2)4 
a 24 m2 ye sc ead Vi 
(a+i)4 +m9)\(% +)-G er 


=(a+0){1-(4 (2 sl 


en: —. = Sae 5 SS Me 


Saas 


eS 


ae SaaS SS 


a 


See oa 


a 


SS SS 


ia 


602 Dr. A. Russell on the Convection of Heat 
Thus 8 has extreme values when 
C= id, one 
and when 
e=—la and y=—=mé, 
Hence we find that 
Bi —By = 2(a+d). 


Substituting this value in (11) we get 


Hada / 22k J 2+) 0...) 


Hence the cooling of the cylinder by the stream of liquid is 
independent of the direction in which the stream impinges ° 
on it. Hor a given area of cross-section and a given tempe- 
rature the cooling power increases with the eccentricity of 
the ellipse, being a minimum for a cylinder having a circular 
cross-section. It is not permissible to apply (14) when the 
liquid is flowing parallel to the minor axis and b/a is a very 
small quantity. In this case, the velocity of the liquid 
round the pointed ends of the ellipse would be very high and 
eddy currents would be formed. 


f. Flat Strip. 


If the solid be a thin strip of metal placed so that its 
length is perpendicular and its surface parallel to the direc- 
tion of flow, we have by (11) 


sokVb ? 
H=4\/ 7 Oy a e+e 


where 0 is the breadth of the strip. This also follows from 
(14). The convection of heat h per square centimetre of the 
surface of the strip per degree of temperature per second is 


given by Spopnt } 
Myf Ty 1). 


If the strip be bent so as to form a hollow cylinder of 
circumference b, we have, by (13) 


paday 2 [sakV 
1. - ah 


= 319 / sh approximately. 


from a Body cooled by a Stream of Flud. 603 


Hence the average convection per square centimetre of 
effective surface is considerably increased. 


| 8. Cylindrical Tube. 
Let the length of the tube be 4, its temperature @, and the 
velocity of the liquid flowing through it V. 
In this case, when the steady state is attained, Poisson’s 
equation (1) gives us 


379106 , 0° soV 30 


oF tr Or Oe kOe 


' As the mathematical formule are complex we shall simplify 
the work by neglecting the conduction of heat in the direc- 
tion of the flow, in which case 


3°0. 190 scV 90 
Or ror —— hk: Ou e 2 ° ° ° (16) 


It is easy to show that the equation * 


eo) 9 ; 
be 5) a ene cos (2n+ L)ma (17) 
wT 9 2n+1 2a a 


makes y, 6, from «= —1/2 to +1/2, 0 from 1/2 to a—1/2, —0, 
from a—l/2 to a+1/2, 0 from a+l1/2 to a+3l/2, and so on 
periodically. 


Let us now consider an infinitely long tube. Take the 
origin at the centre of a portion of it of length J which is 
maintained at temperature @. Let the contiguous portions 
be of length a—/ and be kept at zero temperature, and let 
the portions beyond these be of length / and be at tempera- 
ture —@), and soon. Then by taking afl suttciently great 
we can ensure that the liquid entering the hot portion of the 
tube is practically at zero temperature. 

Writing equation (16) in the form 


0709 100 (2n+1)m? 0A | 
OF 7 r Or (ntl) (mfa) da” ea) 


* Ruseell, ‘ Alternating Currents,’ vol. ii. p. 388. 


604 Dr. A. Russell on the Convection of Heat 


where m’=7soaVjak, we deduce thiat * 


i pa 40, if ber mR ber mr + beimR beimr . al TH 


wr ber? mR + bei? mR Oe 
_ Lherm/3R ber m/3r + beim /3RbeimY3r.. 301 dare 
ak ber? m/3R + bel? m,/3R apg teen 
i JA ae 
ii ber mR bei mr — bei mR ber mr wl. oma 
ber?mR + bei?mR oe ee wee ae 
on Lber m/3R bei mV 3r— beim J 3R ber mi 3r oe ois dmx 
3 ber?m / 3R + bei?m /3R oN Og ie 


i where R is the radius of the tube. 


i This value of @ satisfies (18), and when r=R, 0=0, from 
—I/2 to +1/2, &e., and thus the boundary conditions are 


1 satisfied. ; 
| Hence the loss of heat H per second from the portion of 
the tube from —//2 to +//2 is given by 


41/2 A 
H= 2arRk 09 dx, when r=R, 
—1/2 or 


G! sokVa ah th, te 
= 16R\/ = 6, { Am) sin” 5" 
1 aa ; 37l 
+ 3, 731 m/3R) sin? OF 


how aiehes 


where fll) = % 
_ ber & ber’é + bei & bei’é 
~~ ber + beré 


It will be seen that H is proportional to 6, but except in 
the case when mR is great it is not proportional to 


Rv sokV1. 


I have to thank Mr. H. Savidge for permission to publish 
the table of the values of Z/X given below. This, in addition 


* Russell, Phil. Mag. April 1909, p. 535, 


from a Body cooled by a Stream of Fluid. 605 


to the table * and formule + for Z/X previously published, 
makes (19) a practical formula. 


9. Tables of the Values of the Function Z/X. 


| 


z. ZX. a Z/X. z, | ZX. | 

ees i eee | 

! 3-0 0:5399 44 0:5925 58 06180 | 
3-2 0'5550 46 0:5964 6-0 06211 

3-4 05656 || 48 0-6002 6-2 06210 

3-6 05734 | 50 0-6040 6-4 06267 

$8 | os7s || 52 | oso 66 | 06293 | 

4:0 05842 i) a4 06113 6°8 06317 | 

4-2 05885 | 56 0-6147 7-0 06339 


10. Simplified Formule for Cylindrical Tube. 


In many practical cases m?R?, which equals so VR?/ak 
is a large number, and thus we Ry write 


f@) = 3-H 


When we do this (19) becomes 


sokVa he 2 Sarl ae ,oml | 1 
El = 8 /2R4 / sin? 3 sin? mat Te og 


_ 8ak 7 eae ae 
Bak, {ae 49,800 ie 


If a=2l, so that we have lengths of the tube /, at tempe- 
ratures @, and —@,, separated by lengths / at temperature 
zero, we have 


ae 1 
H=4 /2R4/ a, (1+ ms ir ere ae Oo 


ae 2 t! : 
etd noticing that = nti = 1°6888 approximately, 


we get 


H=1351Ry / S78"! Bere ee CE) 


* Savidge, Phil. Mag. Jan. 1910, p. 56. 
t Russell, Phil. Mag. April 1909, pp. 529 & 532. 


606 Dr. A. Russell on the Convection of Heat 


When the last term can be neglected this becomes 
sokV1 
H=1351Ry /°72™" Goji ss yee. he 
wT 


Similarly when a= 3/1 we get 


ze sokV1 1 \2 1 

SO AVA Si Od st ee aes ere ee 

| 2% any/ T (14 oe ae 
=13-05R, (EY Oy5. «ties | 


and when a = 6/, 
| sokV 
! H=1286Rq/ 27h Oy. 3) a OT 


If the temperature of the liquid entering the tube be zero 
and the temperature of the liquid leaving it be practically 
zero, except at points very close to the tube, we may deduce 
a formula from Boussinesq’s formula (15) for a strip as 
follows: 


asta y mae 6)! ee 


This result is in good agreement with the preceding three 
formule. 


11. Turbulent Flow.) 


It must be carefully noticed that in the above problem we 
have supposed that the particles of water flow in straight 
lines parallel to the axis of the tube. It is known, however, 
that im practice, when the velocity exceeds a certain critical 
value, the flow becomes turbulent and the eddy currents 
cause the particles of liquid to flow in sinuous paths. The 
theory of the convection of heat in this case has been studied 
by Osborne Reynolds *. He states that it is due to two 
eauses. 1. The natural internal diffusion when at rest. 
2. The eddies caused by visible motion which mix the fluid 


* Proc. of the Lit. and Phil. Soc. of Manchester, vol. xiv. p. 9 (1874). 


from a Body cooled by a Stream of Fluid . 607 


up and continually bring fresh particles into contact with 
the surface. In our notation, the formula deduced is 


H = Aé + Bo V8, 


where A and B are constants. 

As the first term is small, H is approximately proportional 
to V. ‘T. BE. Stanton *, who has given an experimental 
verification of Reynolds’s theory, finds that H varies as V” 
where the value of 7 is a little less than unity. 

E. G. Coker ¢ and S. B. Clement have proved that the 
critical velocity at which stream-line motion changes to eddy 
motion varies directly as the viscosity of the liquid and 
inversely as the radius of the tube. 


12. Electric Current required to fuse a Wire. 


Let us suppose that the wire is horizontal with its axis at 
right angles to the direction of the flow of the liquid in 
which it is immersed, and let us suppose that the electric 
current through it is increased very slowly until the wire 
fuses. Let a be the radius in centimetres of the wire which 
we suppose to be cylindrical, C the current in amperes, @ the 
steady temperature corresponding to this current, and p, the 
volume resistivity of the metal at tC. When the steady 
state is attained the heat generated by the current per unit 


length of the wire per second must equal the heat convected. 
Hence, by (12), 


paar Pe os pot a UNE Wh 
0:239C ae) 8 = Bai ia ae GB ) 


C = 7-70 (Alp,)!2(sokV)Vtal® . 2. . (27) 


If 6 be the melting temperature of the metal, we see that 
the fusing current varies as (sckV )"*, and also as the 1-25th 
power of the radius of the wire. This latter result isin good 
agreement with experimental results obtained by Professor 
Schwartz (l. c. ante). In his experiments the wire was 
stretched horizontally in air. ‘he current through it was 
then increased very slowly until the wire melted, the reading 
on the ammeter in the circuit at this instant giving the 
fusing current. Before it melted a vertical stream of air 
was flowing past the wire, the heating of the air by the wire 
causing this convection current. For wires of small diameter 
this current would be approximately constant, and so making 


and thus, 


* Phil. Trans. vol. 190, p. 67 (1897). 
+ Pail. Trans, yol.201, p. 45 (1903). 


‘ 608 Dr. A. Russell on the Convection of [leat 


‘| the assumption that the formule given above for cooling by 
Mh ineompressible fluids may be applied for gases, we see ‘that 
the fusing current varies as the 1:25th power of the radius. 


13. Schwartz's Experimental Results. 


Expressing the fusing current by \a" where \ and n are 
1 constants for a given ” metal, the following results were 
H obtained for A and n. 


‘| Metal. Length of fuse. S.W.G. ee r. nN. 
| | Copper (tinned) .|5 ems. and upwards...|47 to 33/ 1to10) 358 | 1-20 | 
| or ee BR iene. cc. uaeeee : 491 | 126 
| ee ee 76 cms. and upwards.| 43 to 20 147 1-13 
i Pe eh Se ok, 15 cms. S 20 to 7 |10 to 80 239 32 
| Sree est 127 ems. ,, 35to18| 7070] 967 | 1:29 
(| Aluminium ...... 10 ems. i 42 to 20} 2 to 30 640 27 
| In the case of most of the wires placing them vertically 
| did not affect the value of n. Before this paper was published 
I electricians, making the assumption that the heat emitted 
"| per unit surface of the wire was independent of its radius, 
| deduced that n should be 1°5. 


14. Steady Temperature of a Wire carrying an 
Klectric Current. 


If we assume that the volume resistivity of a wire varies 
with temperature according to the law 


Po = Py (1 +28), 
we get by (26), 


sak Va)_-degna reo Ap. qadne CON 
af s/ = —o-2390720 | =o-23007£% (28) 


| and thus @ can be easily computed. The value of C must of 
course be less than the fusing current. 

| Suppose, for example, that the wire is being cooled by a 
stream of ice-cold water. We shall take 

| 

| 

| 


s=o= hb and *=0:0016: 


= 


Hence 


pa oe .  ) 


SE ae +) RS eS ES 


i TN a RE 


from a Body cooled by a Stream of Fluid. 609 
If the rod were of pure copper 
Py = 1:56x10-§, and « = 0-004. 
If, in addition, 
a=0'25 cm. V = 25 ecm/sec. and C= 1600 amperes, 
we readily find from (29) that @ is 11°°3 C, 


15. The Effect on the Convection of Heat from a Cylinder 
of putting a Covering round it *. 


Let a be the radius of the cylinder which we suppose to 
be maintained at a constant temperature 0,, and let b be the 
outer radius of the insulating covering. We shall suppose 
that k,, the thermal conductivity of the insulating covering, 
is large compared with the conductivity k of the cooling 
liquid, so that we can suppose the outer surface of this cover-_ 
ing to be isothermal. 

The equation to the steady mas of heat across the insulating 
covering is 

09 | 

— k,2arr— = constant = H, 

or 

and hence, ; 
| H b 

6,—8 = ak, Beg CU) Re a ae (30) 
where 6) is the temperature of the outer surface of the 
covering. By (12), we find that 


H Wr H b 
ae HOSEN a at at a Col 
8 VsckVb | 2ark, Sea Sy 
Let us now consider how the temperature 6, of the wire 
varies with the thickness b—a of the insulating covering 
when H remains constant. We have 
09; Me H Ts T V trk; 
Ob — 2ark,b3/? 8 V/sokV 
Hence if a be less than 7°k,*/(64sckV), we see that when 
the thickness of the covering is very small 04,/06 i is negative, 
and thus putting on a thin layer of insulating material will 
have the effect of lowering the temperature of the wire. 


When b=77°k,"/(E4sckV) the temperature of the wire has its 
minimum value Boa which is given by 


mk, 
Ca ane a loge 8 ‘/so Tuas a ae (32) 


* Cf. L. Roy, Soc. Int. Elect. Bull. p. 69 (1910). 
Plul. Mag. 8. 6. Vol. 20. No, 118. Oct. 1910. 28 


= 


610 Dr. J. W. Nicholson on the Accelerated 


The following simple experiment illustrates this effect. 
Portions of a piece of thin manganin wire are insulated with 
glass, the rest being left bare. When placed in a current of 
air and heated electrically the bare pieces of wire glow 
brilliantly, but the portions covered by the glass are quite 
dark and are therefore at a much lower temperature. 

In very high tension systems for the electric transmission 
of power the overhead wires are sometimes surrounded with 
eorone which appreciably increase the transmission losses. 
The author has previously suggested that the losses would 
be diminished by insulating the overhead wires with a suitable 
material of high electric strength. The above analysis indi- 
cates that this procedure instead of diminishing the permissible 
current in the wires would actually, in many cases, allow an 
appreciably greater current to be transmitted for the same 
rise of temperature of the wire. 


In conclusion, I have to thank Professor Charles Lees, 
F.R.S., for his kind help in giving me a long list of references 
to papers on this subject. 


LXIII. The Accelerated Motion of an Electrified Sphere. 
By J. W. Nicwouson, M.A., D.Sc.* 


7 HEN a sphere carrying a surface charge is placed in 
a uniform field of electric force at any instant of 
time, it is set into motion under the mechanical action on its 
electrification during the adjustment necessary for the satis- 
faction of the new conditions atits surface. A direct solution 
of the appropriate electromagnetic relations, with a determi- 
nation of the motion ‘of the sphere, has been given by 
Mr. G. W. Walkert for the general case in which the 
sphere is assumed to possess a Newtonian mass in addition 
to its inertia of electrical origin, and in. which the applied 
field of force is small. 

The application of the quasi-stationary principle to accele- 
rated motions has never received formal justification, and in 
addition to certain general considerations tending to throw 
doubt upon its validity for such problems, Walker has 
obtained, in a later paper{, a formula for the transverse 
inertia of a moving sphere which is not in accord with that 


derived by Abraham with the aid of ‘this principle. The 


* Communicated by the Author. 
t Proc. Roy. Soc. 1906, p. 260. 
¢ Phu. Trans. 1910, vol. 210. p. 145. 


Motion of an Electrified Sphere. 611 


method employed is to obtain solutions of the primary electro- 
magnetic relations which satisfy definite surface conditions, 
and neither the relations nor the conditions are dispensed 
with at any stage. After a calculation of the mechanical 
reaction on the ‘sphere has been made, the motion of the 
sphere is worked out by the principles of Newtonian dynamics 
and Walker contends that this method, by its direct ae 
is the one most fitted to yield correct results. With this 
view it seems necessary to agree, and as the method does 
lead to a different formula for the electrical inertia, and, 
moreover, indicates a redistribution of the charge in certain 
cases of motion which is again contrary to the results of the 
quasi-stationary principle, this principle has perhaps been 
pushed too far. Its use is therefore not to be regarded as 
definitely justified in cases of accelerated motion, until its 
exact limits of validity have been examined in a more formal 
manner, and the more direct method seems preferable in 
every way for the solution of special problems. But on the 
other hand, the conditions holding inside a conductor in a 
state of accelerated motion are at present quite unknown, 
and there is no certainty that the evanescence of either the 
tangential electric force or electromagnetic force, conditions 
hitherto used for a perfect conductor, at all represent the 
facts. It is difficult to believe that ious could be no elec- 
trical effect inside a conductor with an acceleration, and all 
that can be done at present apparently is to work out the. 
consequences of various possible assumptions. Thus Walker’s 
results do not necessarily disprove the quasi-stationary: 
principle for small accelerations, and the results of the 
present paper will be found to cast some doubt upon the 
theory that the usual treatment of the perfect conductor is 
still valid when its motion is accelerated. | 

The object of the paper is a brief discussion of the initial 
motion under a small field of electric force, or a small force 
of a purely mechanical nature, of a sphere whose charge is 
initially uniform, and whose mass is purely of electric origin. 
Walker states in his first paper that when the Newtonian 
inertia is zero, the damped harmonic vibration present at the 
beginning = the motion becomes evanescent, and it is 
impossible to satisy all the initial conditions, so that his 
solution fails in this case. The formal deduction of this 
solution as a limiting case from Walker’s formulz is attempted 
in the present paper. 

Prof. A. W. Conway, in a recent paper *, has concluded 
that when a charged sphere without Newtonian mass is 


* Proc. Royal Irish Academy, xxviii. p. 1. 


a 


iH 


612 Dr. J. W. Nicholson on the Accelerated 


placed in a uniform field, it moves in such a way that its 
charge remains uniform. But his investigation does not 
take account of the initial conditions of the motion, and it is 
by no means obvious that the effect of these conditions 
would vanish in the same way as for a sphere with both 
electrical and Newtonian inertia. 

Let € denote the displacement, at time ¢, of the centre of 
a sphere of radius a initially placed in a uniform field of 
electric force Ff of small magnitude, so that F? can be 
neglected. F and €are both measured along the axis of 2. 
The uniform charge initiaily present on the sphere is e, and 
(xz, y, z) denote the coordinates of a point referred to an 
origin instantaneously coinciding with the centre of the 
sphere, 7 being the distance of this point from the origin. 
Then, within the region defined by r=ct+a, Walker shows 
that the components of the electric and magnetic forces are 


given by 
(X, Y,Z)=5 (2, y, 2) +(0,0, 1) fw—S (ty +x’ +8) 


GE Y=59,- 2 0G" 4)" 2 a 


where y denotes y(ct—7), and c is the velocity of radiation. 
eg 
¢€ 

The surface condition is taken to be the continuity of 
(X, Y, Z). Whether this or the more probable (X’, Y’, Z’), 
the electromagnetic force, is to be continuous does not matter 
in the present case, as they only differ by an order F?, The 
surface. condition yields, if €=ct—a, and if the tangential 
component is zero inside, 


ays dey ral CON ae 


x and — are small after the manner of F. 


and the surface density is found to be given by 


ei, ey (oak 
dora = 3 + mo ; —2a*y’’) 0 a ae 


leading to a mechanical force on the sphere of magnitude 


Bn Sg es 1. #0, = (oa 


3a 


Motion of an Electrified Sphere. 613 


along the axis of z, so that if m be the Newtonian mass, 
sti Bee, aay en . 
mE +a x (cha) = ek, Mpa es) 18 (5) 


with initial conditions 
c=f£=0 at é=0. . e ° > ° ° . (6) 


the sphere being initially at rest with ¢ vanishing. 

Oiher conditions may be deduced from the consideration 
that the undisturbed portion of the external medium com- 
mences where r=ct +a, so that y(ci—r) =x‘(ct—r) = 0 
when r=ct+ a, or 


(ane ee eens ate. (7) 
The solution of these equations and conditions is, so far as 
€ is concerned, 


2 Vig 
= 2 eA yon sin { (8+ 4 ct a ; 


3 mac m) 2a" ° 
Leb ug » em Ginn Beene + Amin —m') a? 
: 2 m+m! (m+m')? ce 3° (mm)? * Gg? 
where 
ee ! 4m'\® 
n= 5 +» Asine=—D’, (3+ =) A cos e= —(D'+ 2aB') 
,__ m(2m? + 4mm‘ —m’*) a F pyle mn’ a? 
ES Ont mes Wh ir Naess car 


An error of sign has crept into one of the terms as given 
by Walker, and continues in some of the Jater analysis, 
though not interfering with the general conclusions. The 
value given above has been corrected in this respect. 

The corresponding value of y becomes 


Aeatrtobe 
a a= te} 
m a 


“(ct —1r) = AeW rt a)2e sin { (3 + 


+ A’(ct —r+a)?+ Bi(ct—r+a)+D! 
where : . 
2 ee oe ar ( 
— 4 . m +m . jes e % ° e ° ° ) 
In Walker’s formula (17), p. 264, fcr the value of y after 
the vibrations have subsided, there is an incorrect sign in the 
second term. 
We proceed to an examination of the case in which m is 


SS 


SSS 


614 Dr. J. W. Nicholson on the Accelerated 


very small. It may be shown without difficulty that the 
various constants take the forms 


meat. Om aed ; molt 
peo pa?” ee 
AG ara Cc 2m C 
3 3 3 
m a@kE 5 m \§ aE 
Acme =<— .-—.. A cose ==) ay == 
Qn’ ‘Sie 4° \un’ ce 


c= — ae Bow 22 { eos : a (=y 3(2) sin ; “my 
com a \me a XM Qa \mM 
2 
ee Oe ee 
2m Mm Co “OMA G 
and, except at t=0, € tends to involve the sine and cosine of 
an infinite angle as the Newtonian mass decreases to zero. 
But even in the immediate neighbourhood of the limit m=0, 
it may be shown that this expression continues, like the 
corresponding value of x, to satisfy all the conditions of the 
problem, and moreover, that no other forms can do so. 
Whatever the interpretation to be put upon the sine and 
cosine when m is zero, they cannot exceed unity, so that the 
vibrational term of § will very rapidly disappear on account 
of the damping. A slight departure from the usual condi- 
tion of perfect conductivity in the sphere may perhaps 
remove the indeterminate character of the limit, by preventing 
the argument of the sine and cosine from becoming infinite, 
so that when ¢=0, this argument vanishes, and the initial 
conditions continue to be satisfied with no Newtonian mass 


present. On this supposition, and ¢ vanish with ¢, and the 
initial conditions are satisfied, although the initial acceleration 
of the sphere would be practically infinite. 

The displacement of the sphere may be regarded as a 
superposition of a periodic part upon a part corresponding to 
uniformly accelerated motion, and the damping factor is 
such that the periodic portion is evanescent after an ex- 
tremely small time. The displacement thus tends to the 
form 

Lek) ,..teliat dela? 
Sole! mie Bw 2 a i 

In the formula as given by Walker (p. 268) the sign of 
the second term is positive, and the factor } has been dropped 
in the last term. “i 
_ We proceed to a determination of the surface density on 


Motion of an Electrified Sphere. 615 


the sphere. The coefficient of the zonal harmonic term in 
4aa becomes from (3) 
20 4 


a rel ’ 


and, with a little reduction, it. may be shown that the part 
of ¢ not evanescent on account of m is 


ek lek ot /2 eat 
Heke big 7 eS aa 
m 3m a\m 
and thus by (5) 

3a va 

5 ah — oe (ek — mf) 
3Ba* OW 2 eas ct (m/\? | 

= ee a2) 


so that the surface density is finally given by 
e Len ct (m’\? 
dno =, + Foos@ .¢ i cos. (T) > Sie swe a (13) 


and tends very rapidly to the uniform value belonging to a 
sphere at rest with no applied field, whatever the meaning 
given to the cosine. This conclusion is in accord with that 
of Conway. Thus a sphere with no Newtonian mass must 
move, when placed in an electric field of small intensity, 
without a change in its electrical distribution, if the usual 
conditions for a perfect conductor can continue to be valid. 
The value of o at t=0, before the field has influenced the 
distribution by setting up vibrations, is of course 


L fe | 
o= 7,.( 43 + F cos 8). wis aity(14) 


When Newtonian.mass is present, the surrace density 
soon settles down to the steady value 


=(é eee FoosA) —. (15) 


ot 
Am\a? © m+m 


(Walker’s first result for this case, given in (19) p. 265, is 
corrected in a footnote in the second paper), and for a large 
value of m, gives the ordinary electrostatic formula, as it 
should. | 


616 Dr. J. W. Nicholson on the Accelerated 
Effect of a small Mechanical Force. 


The corresponding solution for a small applied force of 
purely mechanical nature, which we may call G, has been 
given in Walker’s second paper. With the previous nota- 
tion, the primary equations, of which the first expresses the 
vanishing of the tangential electric or electromagnetic force 
(these only differ to the second order) at the surface, 
become 

ay! (ct—a) tax! (ct —a) +x (ct - a) — e =0 


me+ 5 x(a) =G ~ bal es 
with the conditions 
(1) x=x’=0 when the functions have argument —a, 
(2) %=f=0 at t=05,°0 6 2. le 2 


and the solutions are 


. 4m’\i ct — 
x (ct — 1) = Ae rte sin} (3+ ~ ) J +e} 


nm 


= — : eA ctf sin {(3+ a a +e} 
3 mac m/ 2a 
ae les Ani a yy Naa ah 0) 
a eat Ti, eae fri ip cea ae se « « (18) 
where 
ued CRM Am’? ___ eFa?m (2m + 3m’) 
Aosue= Gna’) (3+) Aces aa 
When the Newtonian mass becomes small, 


eFa’m 3 eF a? & ' 


A sin e=—~—~, Acose= =.——> 
em’ y) 9 om!’ 9 


m! 
and on reduction, 
t= a Se | cos fee 7 UF ou ae ue 
em! eS, + 9 ™) sin a(= 
; Dat. Zar 
ax sai(@+ % +3), Oiiwial o i 


satisfying all necessary conditions for values of m tending to 


Motion of an Electrified Sphere. 617 


zero. The formula for the surface density of electrification 
at any time 1s 


eg ee SEG LA een 16) 
E aX 


a2 
But 
; 3a 2: 
x’ (ct—a) = Ne (G—mé), 


and finally, when m is nearly zero, 


é 3G ct pm!\2 
Jo ay oa —p —et/2a by (ae 
Atro= nx cos (1 e ~ 224 agg a ))- : (21) 


az 


A constant surface density (as regards time) is therefore 
speedily established, with a term involving the first zonal 
harmonic. Initially, the value is 


o = e/4rra?, 
as it should be. 

But the infinite acceleration with m=0 again appears, 
although it may be formally shown that these are the only ex- 
pressions capable of satisfying all the hypothetical conditions. 
The motion does not seem, therefore, to be physically likely 
to occur, and the results serve to indicate that an assumption 
of perfect conductivity with the ordinary condition cannot 
readily be justified in an accelerated system, and is of a very 
artificial character. That the electrical motions of the con- 
ductor should be confined to the surface in this case is very 
unlikely, and in the case of a single electron, it is difficult to 
find a physical meaning for the assumption. 

In the more difficult case in which the sphere has a steady 
motion on which a longitudinal or transverse acceleration is 
superposed, a calculation of the electrical inertia on the basis 
of the two usually adopted surface conditions only leads to 
two values which must be regarded as somewhat arbitrary, 
and although one formula may be more supported by, for 
example, the experiments of Kaufmann, than the other, it 
still remains as but one of many perhaps equally likely 
results. The agreement with experiment may indicate that 
the proper vector has been made continuous, but not that it 
is zero inside the conductor. Yet in the present state of the 
theory, it seems necessary to emphasise Walker’s contention 
that the Newtonian type of analysis affords the safest mode 
of attack on the problems of accelerated motion. 

The contracted electron is rejected by Walker as having 
no apparent dynamical foundation, but this may he only 


618 The Accelerated Motion of an Electrified Sphere. 


apparent, and certainly it does not seem possible to dispense 
with the Principle of Relativity and its consequences. 
Moreover, Bucherer’s contracted electron gives a very good 
agreement with Kaufmann’s experiments, and it is desirable 
that a direct mode of analytical treatment of an electron 
which changes its shape, not associated with the quasi- 
stationary principle, should be found, but none has been 
suggested as yet. 

‘here is one combination of a small mechanical force with 
a weak electric field which would give a finite initial accele- 
ration to a sphere whose inertia is electrical only, no electrical 
effect being maintained inside. This combination satisfies 
the condition 


Ga —4eR  O e I 


and the corresponding value of is the limit of 


4 ea® EF (m\? 0, 3. Ee. Ler 4 eF at 
— Be —ct/2a pat od Se a, | a ie ea é 
¢ 3 c*m’ ! “a, < sae ( ) 3m’ 3m’ ¢” (23) 
‘ Lie : 
so that the acceleration at t=0 is 3 lt But it becomes 
m 


infinite afterwards. The surface density remains perma- 
nently equal to 


Lali 
i (4 +F cose) 0 fe oe (24) 
so long at least as ¢ is small. 

In connexion with the question of electrical inertia, the 
investigations of Conway and Walker, starting from the 
same differential equations and surface conditions, lead to 
different values of the transverse inertia, that of Conway 
being identical with Abraham’s expression. A comparison 
of the two methods will be made in a later note, for it seems 
that the formula given by Walker in this case is the only 
possible result of a rigorous analysis applied with the vanish- 
ing of the tangential electromagnetic force as its surface 
condition. 


Trinity College, Cambridge, 
1910, May 28th. 


ee 


| 619 | 


LXIV. On Threefold Emission-Spectra of Solid Aromatic 
Compounds. By Professor E. GOLDSTEIN”. 


OME years ago I observed + that bright, fluorescent, and 
phosphorescent light is emitted by a number of aromatic 
solid compounds —for example, naphthalene, xanthone, anthra- 
cene, &c.—if cathode rays strike on these substances, cooled by 
liquid air to prevent their evaporation and decomposition. 
In this way I was also able to obtain bright-light emission 
from a great many substances, which at an ordinary tem- 
perature are liquid bodies—for example, benzene, the three 
xylenes, benzonitrile, the chinolines, acetophenone, &c. The 
light emitted by these substances gave bright discontinuous 
spectra of a great variety, all consisting of bands of various 
width and intensity. 

Since that time I have extended this research on nearly 
all aromatic substances which I could obtain in any way, and 
have thus obtained about two thousand emission-spectra of 
aromatic substances and of mixtures of such substances with 
other bodies. 

Of course, time does not allow me to give a complete 
report of this work. Here I just want to speak about one 
result of my experiments. 

In the beginning I was satisfied to observe just a single 
spectrum for each substance, because it was thought that 
every substance could emit only one single spectrum. But 
soon I found that the complexity of phenomena is much 
greater than it seemed at first sight. For each substance 
does not show only one spectrum, but, according to the condi- 
tions of the experiment, there may appear three spectra, which 
are quite different from each other, and have no coincident 
maximum. I call these three kinds of spectra respectively 
the inital-spectrum, the chief-spectrum, and the solution- 
spectrum of the substance. 

At the first moment, when cathode-rays fall upon the sub- 
stances, there appears quite alone and bright the spectrum 
which I call the inztial-spectrum. Then the brightness of the 
initial-spectrum diminishes and gets fainter and fainter till 
its density becomes very small; but it never entirely disappears. 
When the initial-spectrum gets fainter, the chief-spectrum at 
the same time appears and grows brighter and brighter. 


* Communicated by the Author. Read at the Winnipeg Meeting of 
the British Association, August 1909. 

+t Verhandl. d. Deutsch. Physik. Ges., vi. p. 156, and vi. p. 185 
(1904), | 


t 
t 
| 

5 Bie 

; i 

hee 

i 

1 ie 

it 

| 

it 
ip | 
{ > 
\) Bi 
if 

(i i) 
' 
. ul 
a! 

h i 
i 


— = 


620 Prof. E. Goldstein on Threefold 


The chief-spectrum is for a great number of substances so 
characteristic that it is possible to recognize the substance in 
this way ata glance and without measuring the wave-lengths, 
just as you can recognize nitrogen by its well-known bands, 
or hydrogen, mercury and helium by their line-spectra, 
This is even the case with isomeric substances; for one is 
able to distinguish ata glance, for instance, the three isomeric 
xylenes or other isomeric aromatic hydrocarbons. The third 
kind of spectra, which is quite different from the two others, 
appears if an aromatic substance is dissolved in any other 
liquid or melted compound and the solidified solution is 
exposed to cathode-rays. 

Now let me just say a few words on the properties of 
each of the three kinds of spectra. 

The chief-spectra always begin from the infra-red, never reach 
the violet end of the visible spectrum, but end about the middle 
part of it in the green or in the blue, sometimes even in the 
yellow. I never observed that a chief-spectrum passes the 
wave-length of A460. The chief-spectra consist of narrow 
channelled bands, which nearly always have their sharper 
boundary toward the violet end of the spectrum. The 
number of the bands varies within a wide range for the 
different substances between a few strips and several dozen. 
The distances between them appear generally irregular. 
The substances, when they send out their chief-spectrum, 
look red or yellow or green, or of any other tint which 
occurs with fluorescent minerals or inorganic salts. On the 
other hand, the light which is emitted during the first 
moments of radiation and belongs to the initial-spectrum is 
--at least, for colourless substances—always blue. The dis- 
continuous initial-spectra of two substances are, like their 
chief-spectra, never quite the same; but as in their general 
appearance they are rather similar to each other, so one cannot 
recognize a substance at a glance by its initial-spectrum as one 
can by the chief-spectrum, but measures of its wave-lengths 
are necessary. The initial-spectra begin always like the 
chief-spectra in the red ; but not cnly reach into the green 
or blue, but go on into the ultra-violet. One type of 
initial-spectra occurring especially frequently invariably 
consists of szz groups of bands. Hach of the six groups 
is formed by the same number of strips at the same 
relative distance and intensity ; and as the relative distance 
of the groups themselves is also not very different—at least 
in the prismatic spectrum—the whole spectrum gives the 
impression of having a very high regularity. Such spectra, 
consisting of six groups, with different wave-lengths for each 


Eimission- Spectra of Solid Aromatic Compounds. 621 


individual substance, are, for example, the initial spectra of 
mesitylenic acid, of metatoluic acid, of the anhydride of 
benzoic acid, of toluene and of its halogen substituted deri- 
vates—and of many other substances, especially of those 
aromatic bodies whose molecules contain a single-ring 
group. 

In the groups which contain two or even more benzene 
rings, and especially in condensed substances, one finds also 
other types of initial-spectra, all extending from red into 
the ultra-violet, which I will not speak of in this short 
report. 

The third spectrum of aromatic compounds is shown in 
very characteristic forms especially by dissolved compounds 
of the condensed type; for instance, by naphthalene and 
most of its derivates. The chief-spectrum of naphthalene 
shows the wave-lengths 


539 (very bright) 589 (very bright) 


DAD 615 (probably a doublet) 
560 630 
573 648 


A539 and 589 mark sharp boundaries on the violet side, the 
other wave-lengths belong to the middle of narrow strips. 

The spectrum of the same naphthalene, if dissolved in mono- 
chlorobenzene (which itself gives only a faint and almost 
continuous spectrum) shows the following wave-lengths (all 
for the middle of the narrow strips) :— 


473 Behe 505 ) rather 517) rather 540) 557) rather 
483 oO 910 f bright 523 f faint 545 { 565 [ faint 
582 faint. 


Beyond this last strip the illuminated ground cannot be 
separated distinctly into strips. ; 

One cannot, however, speak of a single solution-spectrum of 
a body, as the solution-spectrum of the same substance varies 
greatly with the solvent. 

The solution-spectrum of naphthalene, for example, shows 
differences, if the naphthalene has been dissolved in metaxylene 
or in orthoxylene or in paraxylene. Therefore, if one sub- 
stance shows remarkable differences in isomeric solvents, one 
cannot wonder that the solution-spectra of the same substance 
show even much greater differences if more different solvents 
are used; for instance, if we compare the solution-spectra of 
the same substance when dissolved either in a xylene or in 
aniline, pyridine, ethyl-alcohol, and ethyl-ether. 


; ae Set FE ee See? 
ee ae 4 : SS 
: > oe SS BRT = 


| 
if 
a 
4 


pl ee 


622 Emission-Syectra of Solid Aromatic Compounds. 


On the other hand, each condensed compound and _ its 
derivates, even in the case of isomers, shows an individual 
solution-spectrum. The chief-spectrum of the $-bromo- 
naphthalene presents a similar aspect to the chief-spectrum 
of the e-bromonaphthalene. But the solution-spectra of the 
two substances, for example, in monochlorobenzene, are very 
different. The solution-spectrum of the «-substance is of a 
similar type to the solution-spectrum of naphthalene itself, 
presenting only an appearance of a certain regularity by the 
occurrence of some doublets, while the solution-spectrum of 
the 8-form is of a quite different type, and shows a most regular 
structure. It consists of four bands, of quite equal aspect, 
extending from the red into the blue. ach of the four bands 
is formed by five narrow strips, the relative distance and 
intensity of which is quite corresponding in all bands. 

The light of the chief-spectra is fluorescent, and disappears 
at the moment when the cathode-rays stop. 

The light of the solution-spectra is phosphorescent, and very 
often one can see it for some minutes after the discharge 
which produces the cathode-rays is interrupted. 

Only very small quantities of a substance are necessary to 
produce a solution-spectrum bright enough to be remarked 
and to be measured. For example, one can detect in this 
way less than the hundred thousandth part of naphthalene 
dissolved in monochlorobenzene or in methylbenzoicester. 

Of course these phosphorescent solution-spectra are, on the 
other hand, a very sensitive test for the purity of aromatic 
substances, or, what is the same, a very sensitive means 
of detecting very small quantities of admixed foreign aromatic 
substances. And I am sorry to say that, among many 
hundreds of preparations of the best obtainable “ purity,” 
the specimens which did not show very marked signs of 
impurities could be counted on the fingers of one hand, if 
there are any at all. 

I spent much time and money in getting even only very 
small quantities of certain substances really pure, for 
example, diphenyl, indene, carbazol, fluorene, and other con- 
densed compounds, and some of the most famous chemists 
helped me kindly by the best known methods ; but at last I 
had to give up the hope of getting any of these substances 
in pure condition. Until now they have never been -pro- 
duced in a really pure state, and I fear that the same holds 
true tor all other aromatic bodies. 08 


[ 623 ] 


LXV. The Relation between Electromagnetism and Geometry. 
By H. Bareman, Fellow of Trinity College, Cambridge, 
and Leader in Mathematical Physics at the University of 
Manchester*. 


1, ECENT theoretical researches in electromagnetism 

indicate that the science of electromagnetism is 
closely connected with the geometry of a system of spheres. 
According to the generalized form of the principle of 
Huyghens, an electromagnetic disturbance at any point in 
space can be regarded as the resultant of a large number 
of elementary disturbances which are propagated in the form 
of spherical waves. It should be profitable then to study 
the geometrical properties of an aggregate of spherical 
waves travelling inwards or outwards with the velocity of 
light. 

“Two distinct sets of properties must be dealt with. First 
of all we must regard the spheres simply as geometrical 
figures and study the geometrical properties in the usual 
way, and secondly we must consider the relations between 
the different spheres when various numbers are attached 
to each. 

If ct denote the radius of a sphere which is contracting 
with the velocity c, it will have contracted to a point at a 
time ¢ subsequent to the moment at which it was first con- 
templated. Similarly, if it is expanding with the velocity 
of light its radius must have been zero at a time ¢ previous 
to the moment when it was first contemplated. We shall 
say in either case that the sphere is the representative sphere 
of a particle which is at its centre at time + ¢. 

For some purposes it is convenient to study the kinematics 
of a particle when different times are associated with its 
different positions, and for other purposes it is convenient to 
study the geometry of the system of representative spheres. 
The advantage of using the second method is that we may 
study the whole history of a particle by considering its 
chain of representative spheres at a given moment of con- 
templation fT. 

A complex of ©? representative spheres which are related 
to one another in some way will be called a view of the 
universe. It may be replaced by the corresponding system 
of particies if each particle is considered at an appropriate 
time determined by the radius of the representative sphere. 

* Communicated by the Author. 

+ It should be noticed that if a particle is moving with a velocity less 


than that of light, no two of its representative spheres with positive 
radii intersect. 


624 Mr. H. Bateman on the Relation 


It has been shown that the fundamental equations of the 
theory of electrons simply describe the properties of an 
arbitrary view of the universe *. 

We may pass from one view of the universe to another by 
means of a transformation which transforms a representative 
sphere into a corresponding representative sphere +. It has 
been shown that the fundamental equations of the theory of 
electrons are covariant for all transformations of this kind. 

This group of transformations possesses the remarkable 
property that the lines of curvature on the wave surface 
enveloped by a system of representative spheres are trans- 
formed into the lines of curvature on the corresponding 
wave surface. The group of transformations is in fact 
identical with that studied by Sophus Lief. <A particular 
transformation due to Ribaucour § which has been called by 
Laguerre || “Ja transformation par directions réciproques”’ is 
easily seen to be identical with the transformation used by 
Lorentz 9, Larmor **, and Hinstein tf, to pass from the views 
obtained by one observer to the views obtained by another 
observer moving with uniform velocity relative to the first. 

2. The late Russian mathematician Minkowski of Got- 
tingen has made considerable use of a representation in 
which a particle which is at the point (2, y, z) at time ¢ is 
represented by a point whose coordinates are (a, y, , ict) 
in a space of four dimensions ff. 

The group of Lorentzian transformations for which the 
electron equations are covariant is then represented by the 
group of transformations of rectangular axes in the space of 
four dimensions. The more extensive group of spherical 
wave transformations for which the electron equations are 


* See a paper by the author “ On the Transformation of the Electro- 
dynamical Equations,’ Proc. Lond. Math. Soc. (1910). 

{| Asimple transformation may be obtained by increasing or decreasing 
the radii of the spheres by the same amount. Other typical transfor- 
mations are displacements, magnifications, and inversions. 

¢ Mathematische Annalen, vol. v. Géttinger Nachrichten (1871). 

§ Comptes Rendus, t. xx. p. 332 (1870). 

|| Zoid. t. xcii. p. 71 (1881). See also Darboux’s Théorie des Surfaces, 
t. i. p. 253. 

q Usledetdirditn Proceedings (1904). The covariance of Maxwell’s 
equations was established by Voigt ,Géttinger Nachr. 1887, p. 41. 

** /Kther and Matter, 1900. 

+t Annalen der Physik, Bd. xvii. (1905). 

{ft Géttinger Nachrichten, 1908. Physikalische Zeitschrift, 1909, 
pp. 104,216. The transition from Minkowski’s representation to our 
representation gives rise to a very interesting correspondence between 
the spheres in space and the points of a space of four dimensions. This 
correspondence has been studied by Darboux, Annales del Ecole Normale, 
1872. . 


between Electromagnetism and Geometry. 625 


covariant is represented by the group of conformal trans- 
formations of the space of four dimensions *. 

Minkowski, Born f, and Herglotz f endeavour to represent 
the paths of a system of connected particles by means of the 
orthogonal trajectories of a system of «©! hyperplanes in the 
space of four dimensions. The sections cut out on the dif- 
ferent hyperplanes by a tube of orthogonal trajectories may 
be derived from one another by means of displacements (7. e. 
transformations of rectangular axes) in the space of four 
dimensions, and so the corresponding views of the system of 
particles are derived from one another by means of trans- 
formations for which the electron equations are covariant. 

This result may be generalized by considering the ortho- 
gonal trajectories of a system of hyperspheres (or spheres) 
in the space of four dimensions. It has been proved that the 
sections of these hyperspheres (or spheres) by a tube of 
orthogonal trajectories, may be derived from one another 
by conformal transformations of the space of four dimen- 
sions §. The corresponding views of the connected system 
of particles are consequently derived from one another by 
means of transformations for which the electron equations 
are covariant. 

3. The space-time vectors introduced by Minkowski || admit 
of simple representations by means of our representative 
spheres (or spherical waves). 

If we take a particular sphere A as the sphere of obser- 
vation, its relation to a second sphere B may be specified by 
a space-time vector (AB) of the first kind which has the 
effect of displacing the sphere A so that it becomes con- 
centric with B and at the same time of increasing or 
diminishing its radius so that it becomes equal to that of BY. 
The vector, in fact, is exactly analogous to a displacement 
vector from one point to another. 

Now just as there are different physical quantities which 
may be represented by vectors, so there are different phvsical 
vectors which may be specified by means of space-time vectors 


of the first kind. 


* See papers by E. Cunningham and the author, Proc. London Math. 
Soc. 1910. 
t+ Ann. d. Physik, vol. xxx. (1909). Physik. Zeitschr. vol. x. p. 814 
1909). 
t san d, Physik, vol. xxxi. Heft 2 (1910). 
§ This is practically done by Darboux, Legons sur les Systemes 
orthogonaux, Paris, 1898, Ch. II. 
|| Géttinger Nachr. 1908. Phystk. Zettschr. 1909, p. 104. 
“| The components of the displacement of the centre and the change 
in radius may be taken as the four components of the space-time vector. 


Phil. Mag. 8. 6. Vol. 20. No. 118. Oct. 1910. 2T 


626 Mr. H. Bateman on the Relation 


In order to specify a space-time vector of the first kind 
through a given sphere of observation A when the above 
representation is used, it is sufficient to know the position of 
the centre of similitude of the spheres A, B (attention being 
paid to the signs of their radii in determining the choice of 
one out of the two centres of similitude), and a number 
indicating the magnitude of the vector*., If the centre of 
similitude lies within the sphere of observation, the vector is 
said to be temporal, if it lies outside, the vector is said to be 
spacial |. Two vectors are said to be orthogonal | when the 
corresponding centres of similitude are conjugate points with 
regard to the sphere of observation. A vector orthogonal to 
a temporal vector is necessarily spacial, but the converse does 
not hold. 

The relations which two spheres B and C (or a space-time 
vector of the first kind not passing through A) bear to the 
sphere of observation determine a species of space-time vector 
of the second kind containing A. A space-time vector of 
the second kind is specified by six components 


(i, Beye ee 


and the special type at present under consideration 1s 
characterized by the existence of the relation 


HH a, 4 A, 


A more general space-time vector of the second kind may 
be obtained by adding the components of two special space- 
time vectors of the second kind. 

A special space-time vector of the second kind may be 
specified by means of a line, viz. the axis of similitude of the 
spheres A, B, C and a number to indicate the magnitude of 
the vector. ‘The magnitude of the vector may be taken to be 
equal to this number multiplied by the area of the triangle 
PQR, where P, Q, R are the points of contact of a common 
tangent plane of the three spheres A, B, C. 

* The magnitude of the vector may be taken to be equal to this 
number multiplied by the length of a common tangent of the two 
spheres. The magnitude of a vector may vanish although its compo- 
nents do not. 

}~ These terms were introduced by Minkowski, but are defined analyti- 
eally. If (X, Y, Z, cT) are the four components of a space-time vector 
of the first kind, it is said to be temporal or spacial according as 

ce’ T?2X?+ Y?4Z?. 

+ Minkowski uses the word normal. Two vectors whose components 

are (X, Y,Z, cT) (X,, Y,, Z,, cT,) are normal to one another if 


4) Raye ne en 


between Electromagnetism and Geometry. 627 


Two special space-time vectors of the second kind are said 
to be orthogonal when their corresponding lines are polar 
lines with regard to the sphere of observation *. 

A space-time vector of the third kind may be regarded as 
representing the relation of three spheres B, C, D to the 
sphere of observation A. It may be represented by the 
plane of similitude of the four spheres A, B, C, D, and may 
be classified as spacial or temporal according as the plane 
does or does not intersect the sphere. Two space-time vectors 
of the third kind are said to be orthogonal when their repre- 
sentative planes are conjugate with regard to the sphere of 
observation. 

The application of these ideas to electromagnetism depends 
upon the fact that the components of the magnetic induction 
together with the components of the electric force must be 
regarded as the six components of a space-time vector of the 
second kind. In the case of the simplified equation of the 
theory of electrons, the components of the convection current 
together with the volume density of the electricity form the 
four components of a space-time vector of the first kind f. 

The components of the electromagnetic vector potential 
together with the electromagnetic scalar potential form the 
four components of a space-time vector of the third kind ¢. 

The study of the properties of these vectors is facilitated 
by considering integral forms of the type 


He d(y, <)+ Hd(«, «) + H,d(e, y) 
+ H,d(a, t)+ Edy, t) + H,d(z, ¢), 


po,d(y; z, t) +po,d(z, a, t) + pw d(x, ¥y, €) —pd(«, Une) 
A.d(y, 2, t)+A,d(z, x,t) +A,d(a, y, t) —Bd(a, y, 2), 


as in my paper on the transformation of the electrodynamical 
equations. 

It should be remarked that the transformations which can 
be used to transform a -particular electromagnetic field into 
another are not confined to the group of spherical wave 


* A vector of the second kind may be regarded as temporal when its 
representative line meets the sphere of observation in rea! points, and as 
spacial when the line does not meet the sphere in real points. 

{ The principle of the conservation of energy is expressed by the fact 
that the space-time vectors, whose four components are the three com- 
ponent forces and the rate at which work is being done, is normal to tne 
space-time vector of the first kind mentioned above. 

{ There is a reciprocal relation between vectors of the first and third 


kind. 
vied Nil 


628 Relation between Electromagnetism and Geometry. 


transformations. If we denote the velocity of light by unity, 


it may be shown, for instance, that a transformation which is 
such that 


Ada? + dy’? +dz2?— dt” | + plod a v,dy + v,dz— dt}? 
= de? dy hae ae 


is suitable for the purpose provided the vector v whose 
components are (v,, v,, v,) is connected with the components 


(ie Bi) (H, H,, H,) of the electric and magnetic force 
by the relations 


K+ vH, — v,H, =v,( Kv,+ Ky, + Bv,) : 
H vy, + v H y= v,( He + Hye, +H,v,), 
Ps 5 vy +v?=1. 
It can be shown that an expression of the form 
v[ v,dax + vdy +v,dz— dt | 


is an invariant for transformations of this kind and for the 
whole group of spherical wave transformations. I have been 
trying to find a physical interpretation of this vector. 


The University, Manchester, 
June 16th, 1910. 


[Note added Aug. 6th, 1910.] Since the electromagnetic 
equations specify the properties of a view of a set of particles 
and a view is represented by a hypersurface in the four- 
dimensional space, it appears that a transformation from one 
view to another for which the electromagnetic equations are 
covariant need only give a conformal representation of one 
hypersurface on the other, and not necessarily a conformal 
transformation of the whole hyperspace. 

It is possible then that the motion of a connected system 
of particles may be represented by a continuous conformal 
transformation of a hypersurface or, in particular, by a con- 
tinuons deformation without stretching. The path of a particle 
is represented by the successive positions of a point on the 
hypersurface in the successive deformations. The case in 
which the hypersurface becomes torn during the deformation 
is probably irrelevant for physics since a particle corre- 
sponding to a point at which the tear originates would divide 


into two. This case may, however, be of some biological 
interest. 


Ce ee ee ee ee en Tes 


[ 629 } 


LXVI. Molecular Attraction. By J. EH. Mints *. 


N a recent article ¢ on “ The Electric Origin of Molecular 
Attraction,” Mr. W. Sutherland called attention to a 
relation discovered by the author. Mr. Sutherland’s criti- 
cisms of the author’s point of view were largely justified, so 
far as the papers cited by Mr. Sutherland were concerned. 
In later papers by the author, overlooked by Mr. Sutherland, 
the meaning and derivation of the relation were more 
particularly discussed, and the criticisms made could not, I 
think, apply to the views there expressed. The author would 
like therefore to restate the facts and give his own inter- 
pretation of them. 


T. Statement and Experimental Proof of the Fundamental 
Equation. 


The relation under discussion can be expressed in the 
form 


L—E, 

/d—/D 
Here L is the heat of vaporization of one gram of Jiquid. 
Ei, is the energy spent in overcoming the external pressure 
as the liquid vaporizes and expands from the density of the 
liquid d, to the density of the vapour D. L—E, is, there- 
fore, equal to the internal heat of vaporization and is 
designated >. The constant given by the equation is called 
pw’. It is a characteristic constant for any liquid and is not 
affected by changes in temperature. 

The above equation has been tested for thirty-eight 
substances t, anda summary of the results obtained is given 
in the last two papers referred to above. 

I think that the evidence there presented is sufficient to 
justify the conclusion that the equation 


=constant, or N=p'(A/d~</D) wy My) 


ea a dant 
375 SDANGAL an 


represents a new and most ewact law, holding true at all 
temperatures for a/l normal non-assoctated liquids. 


* Communicated by the Author. 

+ Phil. Mag. [6] vol. xvii. p. 664 (1909). 

t Journ. Phys. Chem. Part I. vol. vi. p. 209 (1902); Part IT. 
vol. viii. p. 883 (1904); Part III. vol. viii. p. 593 (1904); Part IV. 
vol. ix. p. 402 (1905); Part V. vol. x. p. 1 (1906); Part VI. vol. xi. 
p. 182 (1907); Part VII. vol. xi. p. 594 (1907); Part VILL. vol. xiii. 
p. 512 (1909). Journ. Amer. Chem. Soc. vol. xxxi. p. 1099 (1909). 


630 Dr. J. E. Mills on 


II. Theoretical Derivation of the Fundamental Equation. 


The equation was deduced theoretically (see sixth paper 
above cited) from certain assumptions which may be stated 
as follows :— 

1. The total energy per se of a molecule must be the same 
in the liquid as in the gaseous state, the temperature being 
the same. If at a given temperature a given weight of gas 
represents more energy than the same weight of the substance 
as a liquid, the extra energy of the gas must be energy of 


> 
position only (assuming no intramolecular change). 


Expressing the above belief in a different form, it may be 
said that the energy necessary to change a liquid into a gas 

must be spent solely in overcoming the external pressure 
and in altering the distance apart of the molecules. (Unless 
the molecule breaks apart also or nears the point of dis- 
ruption.) Hence the internal heat of vaporization must be 
spent solely in overcoming the molecular attraction as the 
molecules move further apart. 

2. The molecular attraction between two molecules varies 
inversely as the square of the distance apart of the 
molecules. 

3. The molecular attraction does not vary with the 
temperature. 

4, The molecules in the liquid and in the gaseous condition 
are evenly distributed throughout the volume occupied by 


them and the number of molecules does not change. 


5. The molecular attractive forces are definite in amount. 
If this attraction is exerted upon another particle, the 
amount of the attraction remaining to be exerted upon other 
particles is diminished by an exactly equivalent amount. 

The above assumptions are, none of them, purely gratuitous 
assumptions made to fit the case in hand. The evidence in 
their favour cannot be given and discussed fully in the 
present paper, but a few comments are warranted by the 
general importance of the assumptions. 

The jirst assumption followed from a study of the kinetie 
theory of gases, the specific heat of gases, and the application 
of the gas law, PV=RT, to solutions. If the gaseous 
pressure was produced by the motion of the molecules and 
a similar pressure (as osmotic pressure) was produced in 
solution, it seemed reasonable to suppose that the osmotic 
pressure was in some way due to an equal molecular motion. 
The molecules of the dissolved substance could not have an 
average kinetic energy of translational motion different from 
the molecules of the solvent. Hence the conclusion that 


Molecular Attraction. 631 


the average translational energy of gaseous and liquid molecules 
of the same temperature must be equal. 

Now a study of the specific heat of gases showed that the 
total energy of a gaseous molecule, exclusive of the energy 
which holds the molecule together and of extraneous forces, 
is proportional to the translational energy. When the causes 
for this relation were considered, it seemed a_ reasonable 
inference that the corresponding portion of the energy of a 
molecule of a liquid would similarly be found to be proportional 
to its translational energy. Therefore the first assumption 
follows *. 

The second assumption was made because all of the 
attractive forces, whose law of variation with the distance is 
known, obey the inverse square law. ‘This is true of 
electrical, magnetic, and gravitational forces. Also the 
intensity of sound, of light, and of heat, vary inversely as 
the square of the distance from the origin. It seemed to 
the author, whatever the nature of the molecular attractive 
force—be it wave-motion or emanation—that the intensity 
of the force must decrease directly in proportion to the 
increase in the surface of the wave or emanation front, and 
since and because this surface increases as the square of its 
distance from the origin, the attractive force must decrease 
proportionately, and therefore obey the inverse square law f. 

The third assumption that the molecular attractive force 
did not vary with the temperature seemed the most natural 
assumption, for none of the other attractive forces, chemical, 
magnetic, electrical, or gravitational, are affected by tempera- 
ture changes so far as is known. 

The fourth assumption that the molecules in the liquid and 
in the gaseous condition are evenly distributed throughout 
the space occupied by them is probably always more or less 
untrue. But if the molecules are shifted from their ideal 
position by reason of the attractive force, the particles would 
gain in kinetic energy exactly so much as they would lose 
in potential energy. It is possible therefore, without error, 

*-The liquid molecules may conceivably possess a “ concealed” 
energy not possessed by the gaseous molecules. If such energy exists 
it is surrendered in proportion to the internal heat of vaporization, and 
its effect is cancelled so far as the conclusions here drawn are con- 
cerned. ‘The evidence upon this point will be discussed in a subsequent 

aper. : 

F 4 I do not intend, however, by this statement to be understood as 
implying that the reason given is the sole reason for the inverse square 
law. The neutralization of the attraction may be another factor tending 
to produce the law. And yet other factors may exist. I am not now 
trying to explain the mechanism. of the attraction or of its neutral- 
ization. Nee, 


632 Dr. J. E. Mills on 


to consider them to be shifted back into their position of 
even distribution ; and the fundamental supposition upon 
which the mathematical work is based is, that the molecules 
of a liquid and the molecules of its vapour have per se the 
same ™ energy when they are in this ideal position of even 
distribution throughout the space occupied by them. 

Except for associated substances or substances undergoing 
decomposition, it is generally believed, and the belief rests 
upon considerable experimental evidence, that the number 
of molecules in the liquid and in the gaseous condition are 
the same. The equation is not true where this condition is 
violated. 

The fifth assumption warrants the closest study. In the 
first paper, when equation 1 was originally deduced, this 
assumption was not expressly made, ‘Ihe deduction of that 
equation contained an error which was later corrected in the 
sixth paper. Attention was called in that paper to the 
following facts :— 

1. The equation, A\=y'( Vd— VD), was true experi- 
mentally. 

2. The above equation followed if a constant mass of 
liquid was taken and the law of the force acting between the 

a) 
particles of the liquid was, force = ~_ , where uw was the 


constant of molecular attraction and was equal to a constant 
times yw’, m was the mass of the attracting particles, and s was 
their distance apart. 

3. If the mass of liquid taken was varied then the same 
law of force between the particles showed that the heat 
required for vaporization oath vary as the 5/3 power of 
the mass. 

4. We know experimentally that the heat required to 
vaporize a liquid varies directly as the mass of the liquid 
taken. 

The question to be determined therefore is, in what way to 

Zay2 


modify the assumed law of the force, namely, force = —- ’ 


in order to obtain the experimentally true equation 
MA= Mp '( /d— VD), 


with either a constant or a variable mass, M. An inspection 
of the factors involved makes it very probable that the 


* See footnote (*), p. 631. 


Melecular Attraction. 633 


trouble is caused by the numerator factor of the force as 
defined in statement 2 above, and not by the denominator, 
Now undoubtedly the molecular attraction is a mutual 
property of the molecules, but it is not necessary to suppose 
that the attraction of one molecule can be indefinitely 
multiplied by the introduction of new molecules into the 
surrounding space. If we assume that the amount of the 
molecular attraction is a constant, and does not vary with 
the total mass of the surrounding molecules, all of the above 
facts can be reconciled at once. From this point of view the 
total attractive force of each molecule is independent of the 
number of molecules and we can write for the law of the force 


constant 
as exerted between two molecules, foree =——;—- But 
§ 


in order to deduce the experimentally true equation (and for 
other reasons) it is convenient to consider the force as being a 
‘function of the mass of the individual molecule and to write 
for the law governing the attractive force of any molecule, 


force = a , where » is a constant and m is the mass of the 


molecule. Now if all of the attractive force is utilized by 

being concentrated upon another molecule we would have 

for the energy necessary to pull the molecules apart from 
distance s, to 59, 


B= ("ym F=pm(>-+ BI Nae apes 


$j So 


For amass of liquid M containing n molecules, and of 
molecular weight m, we have, if v is the volume of the liquid 
and V the volume of the vapour, 


nm=M, male, gale, vt, Va, 


and equation 2 becomes 


ie 
n= he ane (3) 


fer keyed {Oe aie nant 
i ° (<7: oe 
n n n 


This equation gives the energy necessary to pull two 
molecules from each other during the given expansivn if all 
of the attractive force of one molecule be regarded as con- 
centrated upon the other. The energy necessary to pull n 


634 Dr. J. E. Mills on 


molecules from each other is simply n times as great, or 
probably with more exactness n/2 times as great, and we 
have 


SER A M = = 
an (Vd VD) = x= ( Vd—VD).. @) 


If p= vm! (or 2'V/mp’), this equation reduces very 
simply to : fe 
MA=Mp( /d— 7D), °) ee 


which is the law that we have above shown to be experi- 
mentally true. 


ITI. Statement of the Fundamental Equation in a 
Simpler Form. 


While it seems to the author that all of the above 
assumptions are conditions that are probably fulfilled if the 
equation 


MA=Myp!( ‘Vd— VD) 


is true, and I have shown that it is true, I do not mean at 
all to say that the equation as stated really represents all of 
those conditions. The equation rests upon those conditions 
and was derived logically trom them, but the meaning of the 
equation itself is more restricted. Taking into consideration 
the theory by which the equation was derived, it is certainly 
probable that the equation will represent under all circum- 
stances, the temperature remaining constant during the 
expansion, the work done against the force of molecular 
attraction in moving molecules further apart. Now, the 
further the molecules are moved apart the less becomes 
the value of D, and D will finally become zerc when the 
molecules have been moved an infinite distance apart. 
Making, therefore, D equal to zero, and remembering that 
the distance apart of the molecules, s, is proportional to 


+=, where n is the number of molecules and is therefore 
Vv nd 
a constant, we can write 

A, s=constant, . . . -. a 


as the very simple form for the law under discussion. This 
statement means simply this :— 

In any normal substance the internal heat given out as the 
molecules approach each other, mulliplied by the distance apart 
of the molecules, is equal to a constant. — 


Molecular Attraction. 635 


The equation 
L—E, 
ig VD 
and the above statement, are true, because the molecular 
attraction varies inversely as the square of the distance apart 
of the attracting particles and becuuse the total amount of 
attractive force possessed by a molecule ts a constant. 

The author believes that the above statement and italicized 
sentences express the physical reality represented by the 
equation under discussion, 

N=p'( Vd— VD). 
The true nature of the attractive forces is a subject which I 
will not attempt to discuss in the present paper. 

Regarding the errors cited by Mr. Sutherland contained 

in the earlier papers, I wou!d say that I have never supposed 
the molecular culiesive force and the attraction of gravitation 
to be identical in the sense attributed to me by Mr. Sutherland. 
In the first paper, p. 230, I state in italics, “the molecular 
attraction appears to resemble the attraction of gravitation in 
that tt varies inverse/y as the square of the distance apart of 
the attracting molecules and does not vary with the temperature. 
It differs from the attraction of gravity in being determined 
primarily by the construction of the molecule and not by its 
mass.” I have never receded from the above view, and by 
the statements made in the second and fifth papers, that the 
molecular force obeyed the law of gravitation, I did not 
intend to imply that the constant factor of the force was 
identical in the two cases. I still believe that all attractive 
forces may be identical in origin and character and obey the 
same general law, but of course the constant factors of the 
forces, in the usual sense of that term, are totally different. 
The statements as I made them were misleading, I admit. 

I did not at first understand the fact that the law of 
gravitation extended to the molecular attractive force, made 
the heat of vaporization vary, not as the mass, but as the 
5/3 power of the mass. A realization of this fact and its 
consequences caused me to publish, in the sixth paperalready | 
cited (1907), a full discussion of the derivation of the 
equation under discussion, and to express a belief which I 
had long held, namely, that the numerator factor of Newton’s 
law of gravitation needed modification. A further statement 
of my views upon this subject will shortly be published. 


Camden, S. C. 
June 7th, 1910. 


= constant, 


Ee SS Ee eS 


a 


eS SoS 


T= = 


ae = 


P6864) 4 


LXVII. The Series Spectrum of Mercury. By S. R. Miner, 
D.Se., Lecturer in Physics, University of Sheffield *. 


A? we pass up from group to group of the elements in 
the periodic table, the lines in their spectra connected 
by series relations become in general less and less marked. _ 
Thus, while the spectra of the alkalies show on the average 
some seven or eight of the lines in each series, those of the 
alkaline earths show only the first three or four of the lines. 
In mercury, Kayser and Runge have observed only three 
complete members of the triplets of which the sharp and the 
diffuse series of this element are composed, and the principal 
series was entirely unknown until last year, when Paschen 
discovered the first three members of it in the ultra-red. 

In taking some photographs of the spectrum of the 
mercury arc zn vacuo recently, I was struck by the almost 
complete absence of a visible background to the spectrum 
which it showed. In the spectrum of the are in air a limit 
is set to the faintness of lines which can be observed by the 
brightness of the continuous spectrum which is always 
present, but trial showed that with the arc zn vacuo it was 
possible to give exposures over 50 times the normal without 
any background making itself evident. Photographs taken 
with these long exposures showed a great many lines which 
have not been previously observed, but the chief interest 
about them was that the lines which form the continuation 
of the various series of mercury were very strikingly 
developed. 7 

The mercury arc used was a very simple home-made 
apparatus similar to that described by Pfund f ; the are was 
about 5 cm. long, and was worked at 4 amp., 15 volts. 
With a Hilger single-prism quartz spectrograph, with which 
the normal exposure was about 30 seconds, an exposure of 
half an hour showed the lines of the diffuse series up to 
m=16, and those of the sharp to m=14. Traces of still 
higher lines could be seen, but they did not come out any 
better with a longer exposure of two hours, as a continuous 
background then appeared sufficiently strong to mask the 
continuation of the series. With a suitable instrument of 
higher dispersion (which would diminish the continuous 
spectrum) it would doubtless be possible to extend the 
series further, but attempts with a two-prism calcite spectro- 
graph were not very successful through the much greater 


* Communicated by the Author. 
| Astrophys. Journ, xxvii. p. 299 (1908). 


The Series Spectrum of Mercury. 637 


absorption than quartz which calcite exerts in this region 
(X 2500). An exposure of eight hours through the calcite 
prisms gave no greater effect than half an hour through the 
uartz. 

The following are the wave-lengths and frequencies of the 
lines of the series obtained by comparison with the iron are 
(Kayser, B. A. Report, 1891). The first components only of 
the triplets are given: the series formed by the two other 
components are also well developed, but the higher lines in 
them lie beyond the region in which quartz is fairly trans- 
parent, and require very long exposures to bring them out : 


Diffuse Series. Sharp Series. 
| | a 
| m. | 2X (air). m(vac.). || — 2 (air). m(vac.). | 
| ae 
| | (3663-46 272890) | | | 
| 63-05 292'1 | peat 
| eh eee 546097 | 183068 
| 50°31 387°3 | 
3027 66 330194) || / | 
25°79 039°8| | ten” 
| 3 | Bae 0633} | 8341-70 299165 | 
| | 21-68 0848) | | 
| 4%. 280369 | 896870 | e551 | sai723 | 
| oF... 2699-74 | 270299 | «275088 «=| = 3622386 
| 6......| 2680-92 378689 267520" 873605 
| Tone} 2608°10 | 384045 | 262537 | 38078-7 | 
ore...) 257834 38773°2 | 2593-43 | 38547°7 
Bee....| 2561-15" 390335 | 257185 | 38871-0 
10 ....| 254851 39227°1 2556°36 391066 
as ares fo" ieee 254509 | 392798 
| $a). 2531-74 Ey k! ee ae eee yn RNC ee | 
| 132...) 2525-90 895782 | 252047 895223 
lk al 252127 | 396509 | 2524-48 396005 
OT Re rege sy erie 397092 | | 
bt TGsgcas, 2514-48 | | 
| 


39758-0 


* Kayser, Spectroscopie, vol. ii. p. 545. 
+ Hidden by the over-exposed strong line, \ 2536-7. 
} Kayser and Runge give a weak line 252480 (B. A. Rep. 1892). 


An exposure of two hours with the quartz prism gave also 
a series of converging lines which was evidently a con- 
tinuation of the principal series which starts in the ultra-red. 


638 Dr. S. R. Milner on the 


Their wave-lengths were measured on the calcite spectro- 
graph. With four hours’ exposure the series down to m=16 
was well developed, traces of several further lines may be 
seen, and on some of the negatives a faint indication of what 
may possibly be the limit. There is a slight diminution in 
density just beyond the point where the limit should 
theoretically be, but it is too faint to speak with certainty 
of it. 


Principal Series. 


| 

mM. Xd (air). 2 (vac.). | 

ee 
ea { 53543 18671°5 
5316-0 188060 
Es br { 5120°93 19522°4 
a 5102°85 19591:5 
eo Ws: { ee 20069°7 
8 { 4884.0 204443 
ae i 4884°0 20469°4 
= ae 4826°36 20713°9 
PGs... 4781-07 20910°1 
re 4747-72 21057-0 
ae 4722-09 21171:2 
i oa 4701°96 21261°9 
ic eee 468486 213395 
is ae 4672°38 21396°5 
PGe:t.. 4661:29 21447°4 


The principal series should theoretically consist of triplets 
converging to a common limit; the higher lines are all very 
diffuse, they show distinct evidence of broadening from 
m=12 to m=9, m=8, 6,5 are double, m=7 is possibly 
double also, as the smaller wave-length component may be 
swamped by the spreading out of the over-exposed strong 
line > 4959-7 close to it. 25354 (m=5) shows a considerable 
broadening towards the red, which may be due to the partial 
separation of a third line of the triplet ; on the other hand, 
Kayser and Runge give a faint nebulous Jine 5365°25 which 
may be the cause of it. Beyond this point it is difficult to 
identify the series lines. Paschen from his observations in 


j 


a 


Series Spectrum of Mercury. 639 


the ultra-red gave the following as probably forming the 
series * :— 


m. (air). m (vac.). 
9 12071°32 8281-937 
Le he 11288716 8856°527 
3 { 7082-72 1411505 

one 6907 93 14472-20 
5890:05 16973°17 

4 ie 5859-59 17061°38 
580418 1722428 


Another peculiar feature which is brought out by these 
photographs with long exposures may also be remarked on. 
It is the extraordinary complexity of the strong lines which 
form the earlier members of the diffuse series. Kayser and 
Runge observed these to be composite and to consist of 
several components, but with these long exposures in some 
cases over a score of faint satellites show on each side of the 


main line. Thus in 13132 a space of 50 A. units on each 
side of the main line is closely filled with them. 


This development of the series spectrum makes the series 
of mercury in one respect more complete than those of other 
elements. In other elements, single series may be more 
extensively developed it is true: thus in hydrogen the diffuse 
series has been observed down to m=31, but the principal 
series is unknown except for the first line. In sodium, Wood 
has recently observed in the absorption spectrum the principal 
series down to practically the theoretical limit, but only 
seven terms of the sharp and diffuse series are knownf. 


* Ann. d. Physik, xxix. p. 662 (1909). The choice for m=4 does 
not fit in well with the higher lines of the series, as judged by the 
frequency differences of their components. Below are given the wave- 
lengths of all the lines observed in this region of the photographs which 
are not given by Kayser and Runge :— 


| | 


Wave-length.| Intensity. ! Wave-length. | Intensity. 
ree aie 5 | aes / 2a 
6907°9 3 5872°24 1 
67050 2 || 5859-63 2 
623455 6 | 567614 2 
6123°62 2 | 5025'80 3 
6072°71 1 | 4995°91 ] 


+ Zickendraht has, however, recently extended the number to 12. 
Ann. d. Phys. xxxi. p. 249 (1910). 


640 Dr. 8S. R. Milner on the 


In mercury all three series are equally well developed and 
extensively so. 

This fact is of special interest because it allows an accurate 
test to be made of one of Rydberg’s empirical laws connecting 
the different series with each other. The law in question 
runs as follows :—‘The difference of the frequency of the 
convergence limit of the principal series and that of the 
common limit of the sharp and diffuse series is equal to the 
frequency of the first line of the sharp series.” ‘This law is 
known to be very approximately true, but there must always 
remain a certain amount of doubt about its absolute truth so 
long as any one of the series is represented by only a few 
terms all of them remote from the limit. The limit in such 
acase can only be determined from an empirical equation 
designed to represent these terms; at the best this can be 
but an imperfect representation of them, and the value of the 
limit will to a certain extent depend on the particular form 
of the equation adopted. But this difficulty does not arise 
when.so many lines of the series have been measured that 
the last members of them are quite close to the limit. The 
extrapolation to the limit itself is then a small one, and any 
approximate equation will determine it accurately. 

Thus if we apply to the principal series Rydberg’s equation 
in the form 


P(m) = P(%o)—N/(m + :90845)? 


we obtain the following values of the limit, P(~« ), for each 
line of the series :— 


m. P(o)—P(m). P(e ). 
eins: 7179°5 21651°7 
AT ( 25 4552°1 217764 
Bt iis: 3141‘7 218132 
Olas 2298°0 20°4 
Got: 1753°6 23°3 
SE sK. 1382:0 26°3 
Dive 11171 31:0 
1028 921°8 31:9 
1h Beer 773-4 30°4 
Lo eee 658'3 29°5 
eae ee 567°0 28'9 
Nee ae 493°3 328 
LG eae 433-2 29°7 
AD Zhe 383'4 30°8 
PE OEGR yl — Vries t P(m) = 17224. 


{ Lower frequency component. 


Series Spectrum of Mercury. 641 


These values are not constant, they undergo on the whole 
a regular progression with m, but if each value be plotted 
against its distance P(~ )—P(m) from the limit, the extra- 
polation of this curve to the axis of zero P(# )—P(m) will 
give an accurate value for the limit, even when the formula 
itself is quite out in the representation of the lower lines. 
The extrapolation by a curve is too great unless the lines 
have been measured close up to the limit, but with the 
sixteenth line given it is both easy and accurate, and gives 


P(co) = 2183245. 


The diffuse series was worked out in a similar way except 
that a modified formula * which gives a closer representation 
of the lines was used : 


| . ‘178601 \? 
D(m) = D(w)—B |(™+ 981485— ee). 


The limit D(o ) calculated for each line of the series from 
the above formula is shown below :— | 


Mm. D( ). mM. D(w ye 
i. 40141-0 yi rest ny ea rnaay 
Sam 126-2 ian 4019-2 
a. 35°7 1! 40:3 
re 35:8 ae 40:3 
Bes: 33:8 fae 39-2 
one 38:3 Tine 38:8 
ee 39'3 


Extrapolation is here almost unnecessary, and the value of 


the limit is 
D(coo ) = 40139°6. 


This, as may be judged from the table, is probably accurate 
to less than a unit. The principal limit is not quite so 
accurately determined owing to the smaller dispersion of 
the prisms in this region of the spectrum. ‘ 

The difference between these limits is 18307:1. The 
frequency of the first line of the sharp series (A 5460°97) is 
18306°8. The agreement is unmistakable. Thus the case 


* This is one of the variations of that used by Hicks, Phil. Trans 
vol. ccx. p. 57 (1910). 
Phil. Mag.8. 6. Vol. 20. No. 118. Oct. 1910. 2U 


642 Prof. J. H. Jeans on the Analysis of 


of mercury with its extensive number of series lines forms 
strong evidence in favour of the absolute accuracy of 
Rydberg’s law. 


Nore added July 1910 :—Since the above was in press I 
have remeasured the first few lines of the Principal Series, 
using a small plane grating on the spectrograph in place of 
the caleite prisms, with the. following results :— 


m. \ (air). mM. \ (air), 
536603 ; ” 
5 Sy Wu ( 5353:96 i eteores 4980 78 
| 581695 Beni 4889-79 
6 { 5120°84 
MEN 5102°49 9 ...... 4826°82 


15 hours exposure was required ; the lines given were 
quite sharp, and their values are I think now correct to 
about ‘1 unit; the remaining lines were not sufficiently 
developed to distinguish. 

The new lines in the footnote on p. 639 were also (except 
the first two) remeasured with the grating, and the values 
there given are the corrected ones. Most of these lines have 
been observed by Stiles, Astrophys. Journal, vol. xxx. p. 48 
(1909). 


LXVIII. On the Analysis of the Radiation from Electron 
Orbits. By J. H. Jeans, WA., F.RS.* 


t. ii the present paper an attempt is made to examine the 

nature of the radiation which would be emitted by 
electrons describing orbits about various centres of force and 
in fields of force of various kinds, with a view to collecting 
evidence as to whether black-body radiation can be inter- 
preted as radiation emitted in this way. 

In a previous paperf a proof has been given that if 
radiation can be explained in this way, the orbits must be 
described about centres of force varying as the inverse cube 
of the distance. The present investigation confirms this 
result in an independent manner, and tests how far such 
radiation would be in agreement with that observed experi- 
mentally. 


If \Eydn is the partition of radiant energy in matter, the 


* Communicated by the Author. 


+ “On the Motion of Electrons in Solids,” Phil. Mag. [6] xvii. p. 773 
and xviii. p. 209 (1909). 


the Radiation from Electron Orbits. 643 


rate of absorption per unit volume is, as Sir J. J. Thomson * 
has shown 
J 4me, V?Enda, 
where c) is the conductivity for waves of frequency equal to 
that of light of wave-length ». For this to be equal to the 
emission, say {F\a per unit volume, we must have :— 
Dyce een dy ds © (1) 

Since Ey, is the same for all kinds of matter, it follows that 
the ratio of c, to F, must be the same—the conductivity stands 
in a constant ratio to the emission in the interior of a solid, 
in spite of the fact that c, varies greatly from one substance 
to another. It seems legitimate to draw the inference (at 
any rate as a working hypothesis) that the mechanism of 
emission must be the same as that of absorption. 

For long waves, the mechanism of absorption is almost 
certainly to be found in the motion of free electrons, and the 
supposition that this is also the mechanism of emission is 
known to lead to results which are in agreement with 
experiment. 

For short waves, there is less certainty as to the mechanism 
of absorption. It seems probable that other agencies, such 
for instance as various types of resonance, contribute some- 
thing to the absorption of light of short wave-length, and it 
may be that at the wave-lengths with which we are primarily 
concerned this contribution may greatly preponderate over 
the original contribution from the motion of free electrons. 
If sv, we should have to look for the origin of emission as 
weli as of absorption in a motion subject to resonance, such 
for instance as the motion of electrons in small closed orbits. 

It follows that we have to analyse the radiation from both 
closed and open types of orbits, although naturally only those 
closed orbits need be considered in which the motion is stable 
for all possible displacements. 


General law pr-". 


2. Consider first the motion of an electron in a single orbit 
described under a law of force wr-”. The equations of 
motion are: 


— 70? = pro”, meteem ae heer 3S. AOR) 
Pe em; (3) 


where H, the moment of momentum, is a constant of the 
* Phil. Mag. [6] xiv. p. 223 (1907). 
2 U2 


644 Prof. J. H. Jeans on the Analysis oy 

orbit. Eliminating the time, the equation of the orbit is 
du 
qm te ae se 


where wu as usual is 1/r. If, is the value of wu at the apse 
nearest the origin, integration with respect to u gives : 


du\? 2 
If we now put wu = ucosy and 
Quue-8 
— 0 
oe Hal) 5 ee (6) 
this has the integral : 
eh sin vy dy 
y =| sae) ee 
From (3), the time is given by 


‘i do 1 ' sin y dy : 
a cos” x fsin? y + a(1—cos”-! y) 14" (8) 


Hy’ Ha 
The components of acceleration are 
pr” cos @, pr-" sin @, 


from which the radiation can be written down. 
Resolved into its constituents by Fourier’s Theorem, the 
radiation from the complete orbit is 


Qe {essa 9 
anVJ, 6 Pr > le a) oe 


where 
I = fur cos@cosptdt. . . . . (10) 
J=|pr-“sin@sinpidi, . . , 7 ip 
in which the limits are t= —o to t=+00 for an open orbit, 


and are taken through a complete revolution if the orbit is 
closed. On substituting the values of @ and ¢ from equations 
(7) and (8), and effecting the integrations with “respect 
to x, we obtain : 


Le pur AID (a, p/Hu?), 
l= pun? A! (a, plHu), 


the Radiation from Electron Orbits. 645 


in which ®, ®’ are functions of which the form is not at 
present required. If ©?+” be denoted by V, the integrand 
of expression (9) for the complete radiation becomes: 


ea we ie eee , .  (12) 
Let ¢ be the velocity at the apse, so that Huy =c,. The 
value of 2 becomes (cf. equation (6) ) 
~ 2p (13) 
ei = @ =) Ee es 


so that « is the ratio of potential to kinetic energy at the 
apse, and expression (12) becomes 


1(n—1)a%eSW(e, pie). « + + + (14) 


_ At this stage the treatments appropriate to open and closed 
orbits diverge. 


Open Orbits. 


3. Open orbits will be described by electrons which are 
free except when in encounters with the centres of forces, 
and the law of distribution of these electrons is known. We 
require first to investigate how many orbits cf any specified 
kind are described per unit of time. The orbit may be 
specified in time by the instant at which the apse is passed, 
this instant being specified analytically by the condition 
0. 

The probability at any instant of finding an electron within 
a given element dw dy dz of volume, having its velocity com- 
ponents within a range dx dy dz is 


NAc dandyde de dy da) twinned a3e(h5) 


where N is the number of electrons per unit volume, and A 
is a constant determined by the condition that expression (15) 
integrated through unit volume shall be equal to N; & has 
its usual meaning in kinetic theory, being given by 
1/2h=RT, and G is twice the total energy of the electron — 
per unit mass, given by 


9 
Ci he ae rl ps et sa sad 


In polar coordinates expression (15) becomes 


NAc-'"¢74 sin? @drd0dbdrdddb. . . (16) 


| 646 Prof. J. H. Jeans on the Analysis o7 
At the apse, from equations (2) and (3), 


De 
a) = 7+ pro” = cu, + pur, 


so that for the electron to come to its apse within an interval 
dt, the value of + at the beginning of the interval must lie 


| 
| within a range (cou +u,) dt of zero. Giving this value to 
| dy in expression (16) we find 
NAeW*"G>* sin? 6 dr dé dd dé db(cu, + pur) dt 

for the number of orbits of a certain type described per 
interval dt. On integrating with respect to @ and @, and 
with respect to all values of @ and ¢ which give a range dey 
to ¢, we obtain 

877N Aer dr ycdley(couo + [uUo) dt, ay a eat 
as the number of electrons which, per time dt, describe orbits 
having 7) and ¢y within ranges d7p and dep. 


The orbit may be more conveniently specified by the con- 
stants Gand H, given by 


We find, in the usual way, that 
dGdH = 2dr de(ce + pur), 
so that expression (17) becomes, omitting the factor dt, 
An’ NAc" HdGdHy » .: 3) oe 


giving the number of orbits per unit time for which G and 
H lie within ranges dG dH. 

4. On multiplying expressions (18) and (14) we obtain 
the total radiation per unit time. Transformed to the variables 
G and a, the new expression becomes 


(n—1)'2 (, Ge? ee oe 
aye |e ian 1 a, p((n—1)z/2u)~*-1(G/(1+4)) i | 
n—3 2 Ps) bane Wes 
x 2m N Ae G na (2qu/n—1) 2-1 ae (. ~m—l(1+a) a i) dG da. 


We obtain the radiation from all possible open orbits on 
integrating this expression from « = 0 to a= +1, and from 


the Radiation from Electron Orbits. 647 
G=0to G=o. The result is of the form 


3n—5 i n+1 pe Es 
= (hm) »—1 f { p(fim) ~ 23 (n—1/2p) il YY (49) 


in which fis a new function of form unknown. 

Since p is proportional to X~* and h to T~? it is clear that 
the emitted radiation would satisfy a ‘‘ displacement-law ” of 
the form 


en emman alc.” « (20) 


Tor natural radiation we must have (n+1)/(2n—2) = 1, 
or n=3, confirming the result previously obtained. 

5. The existence of the displacement-law (20) is, in a sense, 
inconsistent with the displacement-law XT=cons., predicted 
by the second law of Thermodynamics. The law AT =cons., 
however, refers only to a steady state, while the law (20) 
has been derived for the natural state, which (at least in the 
view of the present writer) is not a steady state, as the ether 
has not the amount of vibratory energy required for the 
steady-state condition of equi-partition of energy. 


Law of inverse square: pr~? 


6. The electrons must attract and repel according to the 
law of the inverse square, when at sufficient distance apart. 
There must therefore be some radiation emitted under 
this law. 


In formula mae put n=2, and we obtain 


= (hm)-"f 2up(im)}, . (21) 
so that p enters, ve through the factor p/T but through the 
factor p/T:. This radiation accordingly does not obey Stefan 
and Wien’s law. 


If we replace w by e?/m, and h by oe equation (21) 
becomes 


— ese Te 2om-1(2RT) -3}, 
showing that the emitted radiation is of frequency comparable 
with 4(2RT)??m/e?._ At 300° abs. the value of this expression 
is about 2x10!2; at 600° it is 5x10". Thus radiation 
under the law of the inverse square may exist, but is neces- 
sarily so far in the infra-red as to elude observation. 

This might in itself suggest that we must look for the 
source of natural radiation in collisions of a sharper nature 
than those which occur under the law wr-*. It is not, 
therefore, surprising that our previous analysis has shown 
that the law must be pr-*. 


648 Prof. J. H. Jeans on the Analysis of 
Law of inverse cube: pr-*. 


7. We now put n=3 throughout the foregoing analysis. 
Equations (7) and (8) at once become integrable, and we have 


@ = '\(1+2@) iy, |) ho ey ee 
1 2 
t‘= EEG 5) tan ee (23) 


in which the value of « is now w/H?. If we put a = tan’ 8, 
the value of I becomes (cf. equation (10)) 


I = Gi sin? a| ® cos x cos (x cos B) cos ( pu?G—! cosec B tan x) dx, 


to] 3 


and J is similar except that the last two cosines are replaced 
by sines. 

It is not possible to integrate either I or J or I?+J? in 
finite terms ; each integral can be shown to satisfy a differential 
equation of a known insoluble type. On substituting for I 
and J in the total radiation we obtain F, as a quintuple 
integral. One integration (with respect to G) can be effected, 
but the remaining four integrations cannot be carried out in 
finite terms. Various attempts to evaluate the integral have 
persuaded me that it will not agree with experiment for large 
values of p. 

8. We shall accordingly discuss the form assumed by the 
integrals I and J when p is large. 

Let us put 


K =|? co (ax+tbtany)dy, . . . (24) 
eo 


K'=* cos (—ay-+b ton x) dy, . . oe 


2 


then clearly I and J are the sums of integrals of types K 


and K’. But from equation (24), K is readily seen to be a 


solution of the equation 


oe = K(14 9) | 


and K" is a solution of the same equation with — a replacing a. 
To examine the case of p large, we need only examine the 


— 


the Radiation from Electron Orbits. 649 
case of a/b very small. The equation becomes 
OK _ 
SF = K, 
and its solution is K=Ae~? where A is a function of a only. 


Similarly K'=A'e-*, where A’ is the same function of —a. 
Hence we have 


(27) 


37 
{ cos ay cos (b tan xy) dy = 3(A+A')e™, 


i * sin ay sin (btan y) dy = $(A'—A)e~. 


It follows that I and J are each of the form 
Gif (P)e~PuIG-*coseeB =, (28) 
and that I’?+ J? is of the form 
124 J? = GE(@)e—2PpH*G-*coseeB, | | | (29) 
9. The value of H? is now p cot? B, so that 
HdH = p cot B cosec? B, 


aud expression (18), which gives the number of orbits per 
unit time of given class, becomes 


A4a?N Ae" cot 8 cosec? B dG dp. 
Hence 
87’ NAp aT Jet yh ERAT NL 
i, =~ ay : \ Ge—hG—2pp2G cosec 8 (8) dB dG. 
Integration with respect to G requires the evaluation of 
an integral of the type 


ye ( ge nb nth en OL Pie 40G50) 
ae” 
the required integral being —dy/da. 
It is easily found that y satisfies the differential equation 


Oy 4 
on 6 
of which the solution is y = Aw?K,(ia?), in which v = 4ab, 
and A is a function of a only. Also from equation (30) it is 
clear that ay must be a function of ab only, and therefore a 
function of wz. 


650 Prof, J. H. Jeans on the Analysis of 


Hence A=A,/a, where Ay is a constant. On differentiating 
(30) with respect to a, we have 


ie. 2) 
( e—wG—ha-t GdG=- oH = 2Ay~ K,(i2*). 
0 
When p is large z is large, so that this integral vanishes 
with p large in the same way as e~V* or e—2v (%), Hence on 
integrating with respect to 8 it is found that F, vanishes 
when p is large, through the exponential 


e-2y Aphut) oy 2 (PERT), » |, (81) 
As in equation (1) we have 
Fy = 4repV*Np,) 0. a 


and it is known from experiment that when p is large 
H, vanishes through the exponential 


@ SPIBT (2) ie, 


These results could only be reconciled if we were at liberty 
to suppose that, c, could increase, when p became large, in 
the same way as the exponential e?/RT, But all evidence, 
both theoretical and experimental, indicates that cp, must 
decrease when p becomes large. 

10. It can be seen that the difficulty which has been dis- 
closed by the foregoing analysis is inherent in any theory 
which refers the origin of radiation to orbits in which 
Maxwell’s law of distribution of energy is obeyed. For the 
radiation from a single orbit when p is large must, by a 
general law *, be of the form e—2) in the limit, so 
that on integrating over all orbits we obtain an integral of 


the type (cf. expression (18) 
{e —pf(G)—hmG gg. 


For large values of p the whole value of this integral comes 
from contributions from that value G) of G which makes the 
index of the exponential a minimum ; this is given by 


| pf (Go) = in = ours 


so that Gp is of the form (pT), and the integral becomes 
proportional to 


ePLF(G,)-G.f"(G,)] or e— PE (pT), 


For this to be of the form e~/T required by experiment, 
F\( pT) would have to be of the form c/T, which is of course 
impossible. 

* Phil. Mag. [6] xvii. p. 250. 


the Radiation from Electron Orbits. 651 


11. An alternative is found by assuming that some physical 
agencies are at work which prevent Maxwell’s law from 
becoming established. An approximation to such a state of 
things, which will be seen a posteriori to be sufficiently good 
for our present purpose, will be obtained by supposing all 
the electrons to have exactly the same value of G, this 
being now given by G=3RT/m. The number of orbits of 
given type described per unit time will now be proportional 
simply to H dH, or ($9) to weot Bcosec’ 8, so that when p 
is large the whole radiation F, will be of the limiting form 


\7(B)e —2pp2m/3RT sin 8 dp, 


and this will clearly vanish in the same way as the 
exponential 
o— 2pp?m/3RT 


12. The limiting form just obtained will agree with 
Planck’s law if 


rye an 
hee v, 


where / is Planck’s constant of which the value is 6°5 x 107”. 


Since the value of m is 8 x 10778, it follows that w must be 
3°8 in o.a.s. units. The force exerted on an electron at 
distance 7 is mu/r? or 3x 107*/7?. Thus if the force mp/7r* 
is accompanied by an ordinary electrostatic force +¢?/1?, 
then the latter force will predominate over the former at all 
distances greater than 1°5 x 107* cm. 

The distance of closest approach of an electron to the 
centre of force mp/r? is given by 

Bb Pa eae 

en oe m ”’ 
so that 7=7x10-”/T. At 700° abs. the closest approach 
is 10~’ cm. 

This distance is greater than molecular distances and so is 
much too large to reconcile with the hypothesis that the orbit 
is described entirely under the law wr-?* from a single centre 
of force. Since this is known to be the only type of open 
orbit which can give radiation similar to that observed, it 
appears that the hypothesis that the radiation proceeds from 
electrons describing open orbits is one which must be 
abandoned. The consideration of closed orbits may be 
reserved for a separate paper. 


Cambridge, 
July 23, 1910. 


f) 652. J 


LXIX. The Pianoforte Sounding-Board. 
By G. H. Berry *. 


[Plate XII.] 


N a recent number of this Magazine f it was shown that 

a section of the sounding-board of a pianoforte had a 

natural period of vibration of its own, independent of the 

pitch of the strings upon it, and this natural vibration was 
apparent in all the photographs there reproduced. 

With the particular section used the pitch of the sounding- 
board was roughly 50 and that of the string 261. 

If the pitch of the sounding-board was the same as that 
of the string, it seemed reasonable to expect the sound to be 
considerably re-inforced, and the sounding-board to act as 
the air-column acts, in the usual resonator fixed to the stem 
of a suitable tuning-fork. 

As will be shown, this did not prove to be the case, or at 
least a very important modification is necessary. 


Apparatus. 


The apparatus used was similar to that described in the 
article mentioned, but several improvements have been made. 
The photographic shutter was changed from 1/4 plate size to 
1/1 plate to give a larger aperture. Instead of this shutter 
being opened direct, the tube from the releasing bulb was 
connected to the small shutter. The large shutter, giving 
the actual exposure, was released by means of a small electro- 
magnet. Two dry cells gave the necessary current and the 


circuit was completed by the small shutter, at any desired 


instant up to about 2 seconds, after the hammer had struck 
the strings. 

To determine the speed of the film at the time of exposure, 
a large tuning-fork was used as an interrupter. A wire from 
one prong of the fork touched the surface of mercury in a 
small cup, when the fork was at rest. When the fork was 
vibrating it “made and broke” an electric circuit consisting 
of the fork, mercury cup, two accumulators, the primary of 
an induction-coil, and two contacts, one on either side of the 
shutter. 

The two contacts pressed very lightly on the wings of the 
shutter, and when the wings flew back, on the shutter being 
released, the contacts came together and the current passed 
through the induction-coil, etc., while the shutter was open, 


* Communicated by Prof. Edwin H. Barton, D.Sc., F.R.S.E. 
Tt April 1910, p. 648. 


On the Pianoforte Sounding-Board. 653 


the contacts being insulated again when the wings flew back 
into position. The spark from the secondary of the coil 
marked the film 3°5 cm. behind the exposure line. 

The frequency of the fork was determined by a strobo- 
scopic method. ‘The mean of several results gave 21°50 
vibrations a second. ‘Thus the distance between the centres 
of two time marks on the film represents =< sec. 

The speed of the phonograph drum carrying the film 
was kept as closely as possible the same for all the exposures, 
and was about 69 cm./sec. Thus the whole length of the film 
passed the exposure line in 0°25 sec. 

The magnification on the films in every case is of the 
order 500. 

In the course of the investigation several different sections 
of sounding-board were used. They varied in length and 


Fig. 1. 
DETAILS OF SOUNDING BOARDS. 


u 
hy 
ita 


7 


d 


i 


R 
ZA, a IK 
a Se ee 


PLAN AT B. 


aL 


203 cm. 


}#—— - —— -——_- 
ean sa ae 10°S cm. 


Ae 


Se Ke 


SECTION CL. PLANS. SECTIONS. 


“s6e420 ) 20 do ao Se = no fis so a 


SCALE IN CENTIMETRES 


thickness and therefore had different natural frequencies. A 
drawing of these sections and the way in which they were 
supported is shown in fig. 1. The wood used for the 
sounding-board was that known commercially as “Swiss 
Pine.” The bar at the back of sounding-board was of Spruce 


654 Mr. G. H. Berry on the 


and was slightly curved as is the practice of pianoforte- 
makers. The bridges are of English Beech. The steel strings 
used were of No. 18 gauge and weigh 0°062 grm. per cm. 

The straw forming the connexion between the sounding- 
board and optical lever was in all cases about 5 cm. below 
the bridge. | 


Results. 


On Plate XII. are shown 25 photographs. 

Nos. 1-6 give the natural frequency of the different sections 
of sounding-board used. They were struck with a pianoforte 
hammer. 

In the following table column A gives the distance of the 
point struck from the bottom of the section, column B the 
distance from the right-hand edge, and column C the pitch 
of the strings on the section. 


eee A. B. ee 
cm cm. 
oe 24-0 44 At | 116 
Li A 13:5 3-0 c! «| 261 
‘ieee ae 9:0 25 ce! =| 261 
EE 11:0 25 c! | 261 
Fy RE dS 35°0 50 At | 116 
+ aaa 35:0 50 At | 116 


In every case the wave passes immediately from the right- 
hand edge of the print to the left-hand edge, and in most of 
the prints the waves somewhat overlap. 

In the first six prints the shutter was set to open at the 
instant the hammer struck the board. 

With the exception of No. 6 the amplitude falls off very 
rapidly and the upper partials become more marked as the 
vibration dies away. In practice, no part of the sounding- 
board of a pianoforte has so low a frequency as that showu 
on No. 6. 

In No. 7 the three steel strings were 66°5 cm. long between 
the bridges and were struck at +, from the fixed or lower 
bridge. All the films Nos. 7-25 inclusive were exposed 
about one second after the hammer had struck the strings. 

The opinion as to the tone is that of a pianoforte tuner who 
has an exceptionally good musical ear. He did not know 


Pianoforte Sounding-Board. 655 


for what purpose his opinion was asked, and it may therefore 
be regarded as being without prejudice. 

In No. 8 the frequency of the strings was nearer to the 
natural frequency of the sounding-board than in No. 7. The 
length of strings and point struck were the same. 

It will be noticed that while the 2nd partial was not 
apparent in No. 7 it was very strong in No. 8. This was not 
expected, and it was suggested that the end of the straw, 
where it was gripped by the nut on the optical lever, was 
weak and in some way influenced the result. Accordingly 
the nut was moved along the straw until it gripped as thick 
a part of the tapered end of the straw as was possible. The 
connexion was then quite stiff, but as will be seen from No. 9 
the only effect was to reduce the amplitude of the vibration. 

In No. 10 the strings were of the same length and struck 
at the same point. The pitch of the strings was nearly twice 
that of the sounding-board. The amplitude of the 1st partial 
was greater than in No. 7, and the tone of Nos. 7 and 10 was 
much better than that of Nos. 8 and 9. It is clear that an 
increased resonance is not to be obtained by making the 
frequency of the sounding-board the same as that of the 
strings. The bridge was in all these cases at or near the 
middle of the sounding-board, and the only explanation seems 
to be that the sounding-board divides in half with the bridge 
as a node and when each half has a frequency near the pitch 
of the strings a good note results. This is confirmed by the 
marked resonance of the octave of the strings in Nos. 8 and 0). 
In No. 11 we see that when the frequency of the sounding- 
board is much below half that of the strings, a bad note is 
also the result. The strings were 37 cm. long and 
strack at 4. 

Nos. 12 and 13 were both described as being good. In 
Nos. 14-16 the strings were 48 em. long and struck at 3. 
No. 15 gave an excellent result, the tone was the best of the 
series. The amplitude is very large, the wave regular and 
free from pronounced upper partials. The pitch of the strings 
was very nearly an actave above that of the sounding-board. 

In No. 16 the amplitude becomes rapidly smaller on the 
print. This was due to the strings being slightly out of 
unison. Beats were audible to the ear and also apparent on 
watching the spot of light. If the exposure had been con- 
tinued the amplitude would again have increased. 

Nos. 17-25 were all taken with longer sounding-board 
sections and two copper-covered steel wires taken from the 
corresponding note on a pianoforte. 

To get the distance between the bridges long enough to 


656 On the Pianoforte Sounding-Board. 


take these strings, another frame was made similar in con- 
struction to the previous one but to take only two sections at 
once, and is 187 cm. between the top and bottom beech planks. 
The strings were all 84°7 cm. between the bridges except in 
No. 25 where they were 89'4 cm. All were struck at 4. 

On section K; the bridge, instead of being near the middle, 
was 22 cm. from the top or very nearly } the length of the 
section from that end. 

In No. 17 the 2nd partial is very strong, as would be 
expected from the position of the bridge. 

In No. 18 the two strings had been pulled up three semi- 
tones, which was as much as they would stand. The 2nd 
partial is not quite so strong. 

In No. 19 a longer section Kg was used, but owing to a 
stronger bar there was little difference in the pitch from that 
of Kj. The bridge was 21 cm. from the top, which is less 
than 2 of the whole length 102 cm. 

In No. 20 the bar at back of section Kg had been planed 
down about 5 mm., reducing the natural frequency. The 
tone was worse than in No. 19. The 3rd partial is prominent 
in this case as well as the 2nd. ‘Three times the natural pitch 
of the section is nearly the pitch of the strings. 

In No. 21 the bridge has been moved to the middle of the 
sounding-board. ‘The 2nd partial is not nearly so marked 
but is still easily seen and the tone was much better than that 
of Nos. 17 and 18. | 

The increased length of the sounding-board without a 
corresponding increase in the thickness probably tends to 
encourage the production of the upper partials. 

In Nos. 22-24 the strings had a heavier copper covering 
in order to produce a lower note with the same length of 
string. | 

No. 22 shows a strong 2nd and marked 3rd partial, and 
very little difference in the wave or tone occurred when the 
strings were pulled up a tone. 

In No. 24 the bridge was moved down to the middle of the 
sounding-board. The 2nd and other partials were very 
marked and the tone was not good. 

No. 25 shows the last bass note. The long and weak 
section Kga did not give any good results. 


Conclusions. 
The results of this investigation seem to indicate :— 
(1) The vibrations of the sounding-board, when the bridge 
is near the middle, divide in half with a node at or near the 
middle. 


The Mechanical Vibration of Atoms. 657 


(2) For the middle octaves of the pianoforte, when these 
two halves have a natural pitch near the pitch of the strings 
exciting them, a resonance takes place and a good musical 
tone results. 

(3) For the two tower octaves the statement in (2) does 
not apply. 

(4) Strong 2nd and 3rd partials are detrimental to good 
musical tone. 


14 City Road, London, E.C. 
June 17, 1910. 


LXX. The Mechanical Vibration of Atoms. 
By WituiaM SUTHERLAND*. 


~* account of the electric origin of rigidity and of 
cohesion, both within and without the atom, there is 
no real distinction between the mechanical and the electrical 
vibrations of atoms, but it is convenient to distinguish as 
mechanical vibrations those which can be calculated without 
directly considering the electrical properties of an atom. 

The experimental researches of Rubens and his collaborators, 
Aschkinass, Nichols, and Ladenburg, have carried the mea- 
surements of wave-lengths into extreme regions of the infra-red 
spectrum, where the period of vibration is getting quite 
close to the order of magnitude to be expected from the 
mechanical vibrations of atoms and molecules. The recent 
measurements of wave-lengths by Rubens ana Hollnagel 
for NaCl, KCl, KBr, and KI down to the seventh octave 
below the visible spectrum (Phil. Mag. [6] xix. May 1910, 
p. 761) invite the following brief theoretical investigation. 
Suppose an atom to be replaced by the least cube of the same 
mass and of uniform density that could circumscribe it. 
Let N be the rigidity of the material of this cube, p its 
density, m its mass, m/h==M its ordinary atomic weight or 
mass, and R the length of the edge of the cube, being equal 
to the atomic diameter. Here / is the mass of an atom of 
hydrogen, 1617 x10-™ gramme. The velocity of propaga- 
tion of a shear or simple distortion without change of volume 
through the cube is (N/p)2. The simplest type of vibration 
of the cube would have two opposite faces as middles of 
internodes so that within the atom the fundamental wave- 
length =2R and outside the atom it is X=cr, where c is the 


* Communicated by the Author. 


Phat: Mag. 5. &. Vol. 20. Nov £18. Octe> T910. 2 & 


658 Mr. W. Sutherland on the 


velocity of light through vacuum, or through air nearly, 
and +t is the period of vibration of the cube. But 
T=2R/(N/p)2, so that X=2cR/(N/p)3. 

It is convenient for the mathematical analysis to bring in 
the electric properties of the atom, though we shall not finally 
use them, as will appear immediately. Let the electric 
moment of the atom be denoted by es to be taken as a single 
symbol, and let K be its dielectric capacity. Then in “ The 
Electric Origin of Rigidity and Consequences” (Phil. Mag. 
[6] vii. 1904, p. 417) it is shown that N =27re?s?/3KR®, 
K being introduced to preserve generality. But in “ The 
Nature of Dielectric Capacity ” (Phil. Mag. [6] xix. 1910, 
p. 1) it was found that as regards the relations of the pairs 
of electrons forming the atom K=1. It has also been found 
in my papers on the electric origin of cohesion that between 
atom and atom K=1. I have taken this to be evidence that 
cohesion is due to electric attraction between the electrised 
molecule and its immediate neighbours, the attraction acting 
entirely through the ether in which K=1. Within the 
atom it appears that the constitutive pairs of electrons act 
only upon their immediate neighbours through the ether 
with K=1. It further appears that when we have taken 
account of the pairs of electrons forming matter as a cause 
of dielectric capacity different from 1, we have not to consider 
any other similar agency in the ether. In the formula given 
above for N then we put K=1. In a recent Phil. Mag. 
article on Molecular and Electronic Potential Energy I have 
shown that the cohesional potential energy of unit mass of a 
substance may be written lp? (the Kp? of Laplace or the a/v? 
of van der Waals) where /=4e?s?/m?, the values and laws 
of / having been investigated under M*/ and Mi: in various 
papers of mine on molecular attraction. If then in the 
formula for N we put m?//4 in place of es’, we get N 
expressed in terms of purely mechanical properties of the 
atom, the electrical moment es having been eliminated and 
K reduced to 1. Thus for the velocity of propagation of a 
shear through the atom we obtain the expression (mlp/6)2 and 


X= 2cR/(alp/6)2=6 x 10" x (1617 x 10-7") 3(M/p)3/(alp/b)2 
= 973-6 (M/p)3/(Ip). 
In the following table are gathered all the requisite data 
for computing the wave-lengths of the fundamental mechanical 


vibrations electrically communicated to the sther by the 
atoms of the combined alkali metals and the combined 


Mechanical Vibration of Atoms. 659 


halogens, the wave-lengths being given in the last row in 
terms of w=107*cm. as unit. 


bi. | Na. E. Rb. Cs. F, Cl. Br. Ls 
OM: DE SS 4:6 6:0 73 0-9 2°] 2°7 3°6 
TB oc oc con PO CE Se oe. 56-0 9 19 26 36 
Lo aa ry 23 39 855 153 19 35°4 80” Ie 
1 you oll 20. oe. Toone ot. 1S6° 3:08 3°33 
Re one sox « 19°12 70°69 1510 2862 4403 2942 3516 5059 6039 


To use these results for comparison with the experimental 
ones of Rubens and Hollnagel, I shall form the wave-length 
for NaCl by adding those in the table for Na and Cl, thus 
70°69 + 354°6 =425°3. The next table contains in the first 
row the wave-lengths thus calculated, in the second the 
experimental wave-lengths, and in the third the ratio of the 
calculated to the experimental wave-length. 


NaCl. KCl. KBr. KI. 

SS re 4253 505°6 656°9 7549 

EE re "sda 51:7 63-4 82:3 96°4 
TRAGIGac.3-33 00> 8:23 797 7:98 7-83 


The mean value of the ratio is 8:00. It is rather by chance 
that this ratio comes so exactly to 8, since the separate 
experimental determinations of these large wave-lengths, and 
the data and approximations used in the theoretical calcula- 
tions, do not lead us to expect such exactness at the present. 
But it is sufficiently remarkable that we have found the 
calculated mechanical period of vibration and wave-length to 
be nearly three octaves below the lowest experimental period 
and length yet measured in each case. The theoretical 
fundamental wave-length for LiF is 313°3, which is only 
between one and two octaves below the longest wave measured 
by Rubens and Hollnagel for KI. 

It is necessary to comment on the process of adding the 
wave-length for combined Na to that for combined Cl to 
obtain the wave-length for NaCl. let us consider an 
analogous case in acoustics. Suppose a length of tube J, is 
filled with gas 1, say hydrogen, and with both ends open is 
caused to sound, its period of vibration 7, is 21,/v,, where 
is the velocity of propagation of sound through gas 1. For 
a length J, filled with gas 2, for instance carbon dioxide, we 
have T2=2/,/v2. If now the two tubes were placed so as to 
form a single one of length J, +/, open at both ends, but the 
part J, still filled with gas 1 < /, with 2, and the combined 

2X 2 


660 Prof. Taylor Jones and Mr. Roberts on Musical 


system were sounded, would the period of vibration be 7+ T2? 
In the case where the gases 1 and 2 become identical we 
know that the period of the combined system is obtained 
correctly by adding together the corresponding values of 
7, and t.. I donot know of the general case with two unlike 
gases having been tried. In the radiational case of NaCl, 
KCI], KBr and KI just considered, we have found that the 
periods of the two atoms in each compound have to be added 
together to give the period for the molecule. Now an exactly 
similar result was brought out in my paper on “A New 
Periodic Property of the Elements” (Phil. Mag. [5] xxx.) 
with a correction in “ A Kinetic Theory of Solids” (ibid. 
xxxii.), and further consideration in “The Cause of the 
Structure of Spectra” (bed. [6] ii.). It was shown that the 
atoms of the metallic elements and the molecules of their com- 
pounds at their melting-points have characteristic oscillations. 
The period of oscillation for acompound molecule like NaCl is 
shown to be obtainableas the sum of a period for Na and a period 
for Cl. This fact supports the assumption made above that in 
calculating the mechanical period of vibration and wave- 
length of NaCl we are to add the periods and lengths for Na 
and Cl. Moreover it is interesting to recall that the kine- 
matical explanation which I have offered for the origin of 
Balmer’s formula leads to the consideration of the period of 
each spectral line as the sum of two periods. From the 
calculations given above it appears that ordinary harmonic 
relations are to be expected amongst the wava-lengths of a 
substance in the extreme infra-red. 


Melbourne, June 1910. 


LXXI. Musical Are Oscillations in Coupled Circuits. By 
HK. Taytor Jones, D.Sc., Professor of Physics in the 
University College of North Wales, and Davip K. Ropers, 
B.Se., Isaac Roberts Student of the University College of 
North Wales, Bangor*. | 


sill A Pies) Cele it 


ay a former communicationt a number of photographs 
. were reproduced showing the variation of potential at 
the terminals of the secondary of a pair of coupled circuits 
when the two oscillations of the system are simultaneously 
maintained by a musical arc connected to the primary. In 


* Communicated by the Authors. 
t E. T. Jones and Morris Owen, Phil. Mag, November 1909, p. 713. 


Are Oscillations in Coupled Circuits. 661 


the experiments there described the circuits were so adjusted 
that the frequency of one of the oscillations corresponded 
either to one of the harmonics of the other, or to the perfect 
fifth above it. It was pointed out that in the latter case 
it was necessary tnat the two notes of the system should 
be equally stable in order that the double oscillation curve 
might be produced, and that the note then heard was an 
octave below the lower of the two primaries, being in fact 
their difference tone. 

It was thought desirable to continue these experiments, 
using some of the smaller intervals, in order to find out 
whether the same conditions hold, whether the difference 
tone is produced, and whether the same method of calculating 
the frequencies of the two oscillations also applies to these 
cases. 

The smaller ratios are obtained by diminishing the co- 
efficient of coupling of the two coils, and if we assume as the 
approximate condition of equal stability of the two notes 
L,C, = L,C,*, the value of this coefficient may be calculated 
for any given ratio of the two frequencies. 

If, with the usual notation for the constants of the two 
circuits, we put 


1/L,C,=N,?, 1/L.0,=N,”, M?/L, L,=/?, 


the equation for the two frequencies, n,, 2, becomes 


i 2s By — Be + a“ fh oe N,”)? _ 412N,N} | ; 


Assuming the condition N,=N,, and writing m for the 
ratio of the frequencies, this leads to 


87°n? = 


_ mn’—tI 
nv +1 


Taking as an example m=3/2, as in the former experiments, 
this gives k?='1479. The experimentally determined value 
of k* tor the two coils was °1483. During the singing-are 
experiments the value of k? would be rather less than this, 
owing to the existence of self-inductance in the arc. In 
order to obtain the ratio m=4/3 the value of k? should be, 
according to the above formula, ‘0784. 

There is no doubt, however, that the above condition for 
equal stability, Nj;=No., is only approximate; the value of 
the secondary capacity which makes the two notes equally 
stable depends also to some extent upon the mutual inductance 
of the coils. 

* Cf. E. T. Jones, Phil. Mag. January 1909, p, 41, 


662 Prof. Taylor Jones and Mr. Roberts on Alusical 


The apparatus used in the present experiments was the 
same as that previously described; the mutual inductance 
of the coils was varied by moving the primary along the 
axis of the secondary, and for each position of the primary 
coil the secondary capacity (a variable condenser with oil 
dielectric) was adjusted so that the two notes were equally 
stable. In a certain position of the primary coil the interval 
between the notes, as judged by ear, was a fourth, and if 
then the lower note is sounding, and the areca oth is gradually 
reduced, at a certain point “the note suddenly falls by an 
interval which can be recognized as a twelfth, although the 
deep note thus produced generally dies away rapidly. 

he terminals of the secondary condenser were connected 
to the electrostatic oscillograph, and after a considerable 
number of attempts a photograph was obtained showing the 
wave of potential* in the secondary circuit when this 
difference tone was sounding. The curve in the photograph 
(Pl. XIII. fig. 1) shows the grouping of the waves characteristic 
of simultaneous oscillations of different frequencies. The 
damping is strong in this case, but it was often muca less 
than is shown in the photograph. The curve somewhat 
resembles those which may be produced by simply breaking 
a current in the primary cireuitf. 

The frequency of the groups determined from the photo- 
graph, by comparison with the curve given by the 768 
tuning-fork, was 200°6. 

The constants of the circuits were determined by methods 
which have been fully described by one of us in previous 
papers. In the paper above referred to it was shown that 
in order to calculate correctly the frequencies of singing-are 
oscillations, it is necessary to assume that the are possesses 
self-inductance which must be added to that of the primary 
circuit, the value of the apparent self-inductance of the arc 
depending upon the distance between the carbons. 

In the present case it was found that there was no value 
of J., which made n./n,=4/3, if the resistances of the circuits 
were neglected. After a number of trials the following results 
were calculated. Assuming the value ‘0005 henry for the 
self-inductance of the arc, then N,?= 1°693.10"7, N.?=1°823.107 
C.G.S., k? ="08656. Hence neglecting the resistances, n= (ee 
y= BBS" 7, Ng—n = 209, ‘nsjn ee Wools 

Taking the resistances into account, however, and assuming 
about 4 ohms for the arc, then R;=5 ohms, Rg= 14000 ohms; 

* As explained in previous papers the ordinate of the curve is pro- 


portional to the square of the difference of puns at the terminals of 


the instrument. 
+t Cf. E, T. Jones, Phil. Mag. January 1909, Plate, figs. 5, 6, 7, &. 


es on 


Are Oscillations in Coupled Circuits. 663 


hence, by Drude’s equations*, ng=791'49, ny=591°02, 
Ng—N = 200°46, no/ny= 1°339, 

It is therefore clear that if the arc be assumed to have a 
self-inductance rather less than 0005 henry, and a resistance 
slightly greater than 4 ohms, the ratio of the frequencies of 
the two oscillations will be exactly 4/3, and their difference 
will agree with the observed value ot the group-frequency. 

Pl. XILI. fig. 2 shows the curve obtained with the coils in 
the same relative position but with larger capacities in the 
circuits. This photograph covers the period of change from 
the lower primary note to the difference tone. In this case, 
again assuming ‘0005 henry for ne self-induetance of the 
arc, we find N,?=1-3468.10%, =a ek, kc "08636. 
Hence, neglecting the peel | oh Fo 28, N= ol?" 1A, 
m,—n,=151°14, n/n; =1°3538. Again taking R;=5 ohms, 
R,=14000 ohins, we find by Dréde’s equations n,=690°6, 
m=d17°4, ng—ny=173:2, np/nj=1°3348. 

The frequency of the groups determined from the photo- 
graph is 174°8. In this case also the effect of taking the 
resistances into account is to diminish the ratio of the fre- 
quencies, and possible values can be found for the self- 
inductance and resistance of the are which will make the 
ratio of the frequencies exactly 4/3, and make their difference 
agree with the observed frequency of the groups. : 

“By withdrawing the primary coil to greater distances 
along the axis of the secondary, some of the smaller musical 
intervals, the major third and minor third, may be obtained, 
and deep difference tones sometimes heard; these are, however, 
very unstable and no photographs were obtained for these 
cases. 

By increasing the coupling coefficient to a certain value, 
the two notes may be brought to an interval of an augmented 
fourth, and with a certain value of the secondary capacity 
the two notes may be produced simultaneously. The dif- 
ference tone was not prominent in this case; the impression 
was rather that of the two primary notes sounding together. 
The photograph was easily obtained and is shown in Pl. XIII. 
fig.3. This photograph was obtained with a new oscillograph 
which was at the time arranged for measuring much higher 
potentials, the phosphor- bronze strip being replaced by one 
of steel, and this being under very great tension. This 
accounts for the smallness of the amplitude of the curve. 
The curve was not measured, but it probably eee the 
case n/n, = 7/9. 

Bangor, July 1910. 


* Drude, Ann. der Physik, xiii. p. 584 (1904), 


[ B64 J 


LXXIT. Note on Mr. Bateman’s Paper on Earthquake- 
Waves. By Ropert E. Baynes”*. 


aie the end of his interesting paper on Earth- 
quake-waves in the April number of the Phil. Mag. 
(p. 585), Mr. Bateman connects the times of transit T of the 
first-phase waves to stations at angular distance @ from the 
source, as given by Prof. Milne, by a formula of the type 
T=C+ A@—Bé6, and, by assumption of the relation 
dT/d@=(K/U) cose, where R is the earth’s radius, U the 
speed at the surface, and ¢ the angle of emergence, shows 
that U=R/A if @ is measured in radians, and, by further 
application of Abel’s transformation to the equation of the 
path, that the speed » at distance Ra from the centre is 


given by t Pa 
(1— U*2?/v*)? —sech=! (Ua/v) =(27B/A)loga. . (1) 
The same procedure is of course equally applicable to the 
equation for the time along the path which is given on p. 583 
and which takes the form 
Ta — 2R(o@)de 
U J0 (s—t)3 
on pulling ¢=1—U%e?/v?, (U?a/e*)de = d(t)dt, s=sin’ €; 
the relation assumed above then gives 
T=(7R/2U)(Hs+ 2F), 
where 
E=R/27BU and F=CU/rR+ (A?U?—-R?)/4n BRU, 
and the transformation gives 
ise eb (Pe Pals d 1 
Masih sy, aerials hat ON) gil i 
gee oe (t{—s)? ac 


whence 
log «= Kis—(H+F) tanh-1¢# 
= E(1— U*2?/v*)? — (E+F) sech-!(Ua/v). . (2) 

Comparison of (2) with (1) requires H=A/27B and F=0, 
i,e. U=R/A as before and C=0. The latter result is doubly 
obvious; for T must vanish with @, and the assumed relation 
is not true except with this condition. 

* Communicated by the Author. 


+ In three pleces the factor R has dropped out by a slip and in the 
Table 7/3 should be substituted for y. 


On the Liquid and Gaseous States of Matter. 665 


As a matter of fact Milne’s numbers are better represented 
by Bateman’s formula if 0, 5°69, 45 are substituted for his 
constants *4, 6°1, °5: and these give 9°77 km. per sec. for the 
speed at the surface. 

But, as Bateman’s Table indicates, there is a depth at which 
v has a maximum value: this occurs when U?u?/v?=1—H-3 
in which case log (v/U)=H(tanh-! H-3 — E-3), corresponding 
to sine=E-? with §=(A/B) sin? de, i. e. with the above 
constants to e= 794°, @=154°; and for higher values of ¢ the 
solution will not apply, as there will be no total reflexion 
of the wave. 

For the solution to be applicable throughout the earth and 
with perfect symmetry we must have for the maximum speed 
e=t7, or H=1 (i. 6 B=A/27) with a formula of the type 
T=A0—B@?. The value 11:12 for A with @ expressed in 
radians gives Milne’s results with very fair exactness, and 
thus 9°55 km. per sec. for the speed at the surface, the limiting 
speed at the centre being $e times greater *, where e¢ is the 
base of Naperian logarithms. 


Christ Church, Oxford. 
26 July, 1910. 


LXXIITI. On the Equation of Centinuity of the Liquid and 
Gaseous States of Matter. By R. D. Kurnman, D.Sc., 
B.A., Mackinnon Student of the Royal Society t. 


\HE writer+t has shown that the attraction between 
two pola besides that due to gravitation sepa- 
rated by a distance < is Lo fAS op» 


5 $:(=s 8) (vim)? 


where x, is the distance of separation of the molecules in 
the critical state, T is the temperature and T, the critical 


temperature, and Ba and S4/my, is the sum of the square 


roots of the atomic weights of the atoms of a molecule ; 


* In a problem suggested by Benndorf’s and Herglotz’s important papers 
(‘Science Abstracts ’ for 1907, Nos. 883 and Y85), and set in Jan. £08 
for the Senior Mathematieal Scholarship Examination of the University, 
: asked for the deduction of the relation @=2e--sin 2e from the fancy law 

4(R/U) sine, and also, Abel’s transformation being cited, for the 
oe that the ratio of the speeds at the centre and surface is Ne. 

+ Communicated by the Author. 

t Phil. Mag. May 1910, p. 783: in subsequent references to this paper 
it will be called (a). ‘ . 


eS ee _ 


666 Dr. R. D. Kleeman on the Equation of Continuity 
o(¢, ) is a function whose exact theoretical form is not 


indicated by the investigation except that it has the same 
value for all substances at corresponding states. It was 
found that this function does not vary much with the 
temperature and as a first approximation may be taken 
as constant. Supposing it constant, its value was deter- 
mined and found to be of the order of the magnitude of 
2x 10~ (grm.)(cem.)(sec.)~. The above law of attraction 
is in this paper mace the basis of some equations of con- 
tinuity of the different states of matter. 

Let us suppose that a molecule in the liquid state has the 
same amount of kinetic energy or energy of translation 
as in the gaseous. And let us suppose that the molecules 
in a liquid are in equilibrium between the gas or Boyle 
pressure of the molecules acting in one direction and the 
attraction between the molecules and the external pressure 
acting in the opposite direction. Then, if P, denotes the 
negative pressure due to the attraction of the molecules, 
and p the external pressure and p, the Boyle pressure, we 
have 


Pratp= 71 >= + 


where m denotes the molecular weight of the substance. 
This view of the equilibrium of the molecules in the liquid 
or any other state is now usually adopted by pbysicists, 
principally owing to the work of van der Waals. P, has 
been called the intrinsic pressure of the liquid. 

The Jaw of molecular attraction given at the beginning 
of the paper enables us to obtain a more definite and 
fundamental expression for the intrinsic pressure than that 
obtained from van der Waals’ equation of state. It is 
first of all necessary to make some supposition as to the 
relative distribution of the molecules in a liquid. Let us 
suppose, as we did in a previous paper, the liquid cut into 
equal squares by three sets of imaginary planes, one set 
of which is parallel to the surface, and that the molecules 
are situated at the points of intersection of these planes. The 
attraction of a slab of liquid whose thickness is greater than 
the radius of the sphere of action of a molecule on a molecu!e 
at a distance na, trom the surface is 

U=@O C=O U=OD 2 
(S.) = = = 4 Stu), 


U=—O V=—-O U=-F 


where yy Bee 


3 1 
where for ¢(2z) we have now put 2g, = 8), the factor’, 


of the Liquid and Gasecus States of Matter. 667 


and #{z)(]Wm,)? is the attraction between two molecules, 
an 
z= a{(n+w)?+u?+v'}, 

2, being the distance between two molecules situated on 
the same edge of one of the squares. For a proof of this 
expression see paper (a) p. 791. The attraction of the 
slab of liquid on a cylinder of the liquid of infinite length 
and unit cross-section, standing with one of its bases on the 
surface of the liquid, is therefore 


if: —\,2=% v=0 w=a0 Ln 
= Pain, > i ee dy S ) t (n+w), 


n=l v=—o uw=—a w=0 


ae 
giving the number of molecules lying on a plane cutting the 
cylinder of liquid parallel to the surface of the slab. This 
expression gives the intrinsic pressure of the liquid. On 


brinoing — a factor of — outside the summation sign 1t 
oD at 2 tap) 
a 


may be written 


K(P) (svn) 


é 1/3 1 
ny aia 


and K, is a constant which is the same for all liquids at 


where 


corresponding states. Since b2( =, ) in the expression 
ce 


for the attraction between two molecules varies only slightly 
with the temperature, the value of K, will also vary only 
slightly with the temperature. 


On the assumption. that (=, a) or IK is constant, the 


intrinsic pressure in any given liquid can be calculated. 
The value of K, then becomes equal to 

is _ 46 n=O — Uu=n U=O nmt+w 

ee es ka ca aes 
where 1°66 x 107*°* is the mean valne of K obtained from 
ether and carkon tetrachloride at T.2. The value of the 


* (a) p. 802. 


668 Dr. R. D. Kleeman on the Equation of Continuity 


summation quantity is approximately equal to 2°06 ; and the 
equation for the intrinsic pressure thus becomes 


P,= (Py (= Vm)? 1°66 x 2:06 x 107, 


If K is a function of the temperature only, the equation 
will give the correct value of the intrinsic pressure on sub- 
stituting for 1°66 x 10~* the value of K corresponding to the 
temperature for which the intrinsic pressure is calculated. 

Let us, for example, calculate the intrinsic pressure in 


2T. 
ether at corresponding to which K has been determined. 


The values of m, S\/m,, and pare 74X7-1% 107- oae 


and *6907 respectively, taking the mass of an atom of 


hydrogen as 7°1 x 10-”° erm.” We thus obtain 
P, = 1992 atmos. per cm.? 


Later we will compare this value with that found by a 
different method. 
‘he intrinsic pressure, we have seen, is in general given 


hy 


where K, is a constant which is the same for all liquids at 
corresponding states. Now the writer has shown in a 
previous papert that ° 


p= = Mi/ Pe oh (i/o): 


Per Pc denoting the critical pressure and density and M 
a numerical constant. A comparison of these two equations 
shows that the intrinsic pressures in liquids at corresponding 
states are the same multiple of their critical pressures. 
Since p varies only slightly with the temperature when it 
is low and K, is approximately constant, this multiple will 
be at low temperatures roughly a constant whose value is 
ree 1992 
Pe 36°28 . 
tound for ether at => and 36°28 the critical pressure of ether 
in atmospheres. ” 


== 54°9, using for this calculation the ake Onur, 


* It should be noticed that from the way K was determined it follows 
that P, is independent of the value taken for m. 
¢ Phil. Mag. Dec. 1909, p. 903; and (a) p. 788 


of the Liquid and Gaseous States of Matter. 669 


Tn a previous paper * it was shown that the internal latent 
heat: of evaporation L of a liquid is given by 


d) 4/3 > \4/8 LER 
b= 2{a(2)"-a@)"Fevik.. @ 
where p; and p, denote the densities of the liquid and 
saturated vapour respectively, and A, and A, are constants 
each of which is the same for all liquids at corresponding 
temperatures. This equation is based on the law of attraction 
between molecules, given at the beginning of the paper. The 
equation may be written 


1, = L'—L”, 
where Wee ao ay" (y /m,)" 
m \m 
| 4/3 Le 
and ee A» ("2) (S/m,)’. 
m \m 


From the way the above equation has been cbtained, 
it follows that L’ denotes the internal latent heat of evapora- 
tion of a liquid into a vacuum, and L” the internal latent 
heat of evaporation of the saturated vapour into a vacuum. 
The equation (2) for the intrinsic pressure may now be 
written 


ee CES WEEN GO RFF) oe G4) 


where Ke K, 
A,’ 

and is therefore a constant which is the same for all liquids 

at corresponding states. At low temperatures L” is small in 

comparison with L’', and the above equation may then be 


wriiten Dees rnres 


An equation similar to equation (4) may be very simply 
obtained if we make the supposition that matter does not 
consist of molecules but is evenly distributed in space. Let 
the attraction of a Jarge mass of liquid, making this sup- 
position, on a slab of liquid of unit area and thickness dz at 
a distance z from the surface of the liquid, be (<)dz, in a 
direction at right angles to the surface of the liquid. The 
attraction on a cylinder of unit cross-section and infinite 
length standing with one of its bases on the surface of the 


liquid will therefore be Sah(e). dz; and this is equal to the 
intrinsic pressure. 0 
* (a) pp. 794-795. 


670 =Dr. R. D. Kleeman on the Equation of Continuity 


Let us next obtain the internal heat of evaporation on this 
supposition, making use of the same notation. Suppose a 
thin layer of liquid of unit area and thickness dx is removed 
from the surface of the liquid and distributed in a space of 
infinite extent, the layer being taken so thin that the work done 
in distributing the matter in space is small in comparison with 
the work done in removing the layer. If L, denote the 
internal latent heat per unit volume, 


fds = (iw .W(c).dz= ae) W(z) dz, 
e 0 0 


= L'p = | W(2)d: *, 
Fe 


Comparing this equation with the above expression for the 
intrinsic pressure, we see that 
> ' 
I n = L Ps . ° . . . ° . (5) 
This equation, obtained on the supposition that matter is 
evenly distributed in space, gives on comparing it with 
equation (4) that Ks;=1. Whether this supposition is 
admissible in the above investigation can be tested by 
calculating P,, by equation (5) and comparing it with that 
obtained by equation (2). Thus, in the case of ether at a 
Ge Ws 
temperature of ae equation (5) gives 
754 x 42 X10" x 6907 __ 
10° 


where 75:4 is the internal latent heat of evaporation t in 


ae = 


2187 atmos. per em.?, 


* Stefan has shown (Wied. Ann. xxix. p. 665) that the internal heat 
of evaporation of a molecule is equal to the work done in moving it from 
the interior of the liquid to the surface and then to an infinite distance 
from the liquid. ‘This is true, however, only when matter is not evenly 
distributed 1n space, 2. e. consists of molecules, in which case a molecule 
must be brought from the interior of the liquid to fill up the gap made by 
removing one from the surface. When matter is evenly distributed in 
space, however, we may suppose that during evaporation infinitely thin 
layers of liquid are successively removed from the whole surface of the 
liquid. The radii of the spheres of action of a set of different molecules 
calculated by the writer, Phil. Mag. pp. 840-846, June 1910, on the 
supposition that matter is evenly distributed in space and using Stefan’s 
result, therefore really denote their diameters. 

+ The internal latent heat of evaporation and density data used in this 
paper are taken from a paper by Mills, Journ. of Phys. Chem. vol. viii. 
p- 405 (1904), who has calculated the internal latent heat at different 
temperatures for a number uf substances, using the density and pressure 
data of Ramsay and Young. 


of the Liquid and Guseous States of Matter. 671 


Py te . 

calories per gram at Be which at that temperature may be 
oO 

taken equal to L’. The above value for P, is practically the 

same as that obtained previously by equation (6), viz. 1992. 


Thus we see that Ks or == is independent of the tempe- 


Ay 
rature and equal to unity, or at least approximately so. 
This result throws some light on a very important point. 
So far we have not yet abpamed any information as to 


whether the function ¢, 8) in the expression for the 


attraction between two molecules is a function of the 
temperature or of the distance between the molecules, or 
of both. Referring back to the demonstrations of the 
equations (2) and (3), containing K, and A, respectively, 


it will be seen that if ¢:(= .B) is a function of the 


temperature only, it can at once be taken outside the 

summation and integral signs, and then appears as a factor 

of K, and Ay, which disappears in a or K;. The expression 
| 1 

thus appears to be a temperature function only. This point 

will be discussed at length in a separate paper. 

A general equation of the different states of matter will 
now be developed and some special cases of this general 
equation considered. 

Substituting for the intrinsic pressure in equation (1) 
from equation (2), we ain 


; PEs (a) (evn) = mo 

There is one point which has not been taken account of in 
formulating this equation, to which attention was first drawn 
by van der Waals. When the density of a gas is so great 
that the diameter of the molecules is comparable with their 
distance of separation, the diminution of the mean free path 


- of a molecule on collision owing to its finite size is appre- 


ciable. The pressure is therefore greater than that given by 
Boyle’s law, and according to van der Waals is such as if 
the volume of the gas were smaller than it actually is by 
four times the space actually occupied by the molecules. For v 
we must therefore write (v—b), where 6 is the space occupied 
by the molecules, The effect produced by the molecules 
having finite size is quite large. Thus, consider a liquid at a 
low temperature: the pressure of its saturated vapour or 


672 Dr. R. D. Kieeman on the Eyjuation of Continuity 


external pressure is then small in comparison with the 
intrinsic pressure, and equation (1) becomes 
RT 
Pe = ——3 


mv 
. e e ry RT . 
or the intrinsic pressure would also be given by —— if the 
i MU 
matter obeyed Boyle’s law. For ether this equation gives 
P,, = 240°3 atmos. per cm.? 


This is‘a much smaller value for the intrinsie pressure than 
that obtained from equation (2) or (5), viz. 1992 and 2187 
atmos. per cm.” respectively, and the effect in question is 
therefore quite large in liquids. 

But b is strictly not a constant; the apparent volume of 
two colliding molecules will be influenced by their forces of 
attraction and those of the surrounding molecules, aud con- 
sequently depends to a certain extent on the density of the 
matter. We must therefore write } a function of v and T. 
Equation (1) may then be written 


r+¥(38)(j,) EVM =aG ayer: O 


This is a general equation for any state of matter liquid or 
gaseous, for the same conditions of equilibrium apply to the 
gaseous as to the liquid state. 


We have obtained some evidence that > & i : B) or K, is 


a function of the temperature only, which must be such that 
its value is the same for all liquids at corresponding states. 
Let us first consider the equation taking y3(v, T) a constant 
b and K, a function of the temperature only. Subsiituting 


~ for v the equation may be written 


RT 
1 OED cs 
oie Lae one 
Ps wihcal beta eae iA as 
or 
RT 
bp st Fiat : p ; : 
UO eget cs) apa 6 « 6 
oO b (on + bA, o bA; P] ( ) 
where om 
2 
o=p® and A,;=K, (= v7) 
mm" 


of the Liquid and Gaseous States of Matter. 673 


According to Descartes’s Rule of Signs the maximum 
number of positive roots that ¢ in equation (7) can have is 
three, and the maximum number v in the original equation 
can have is therefore also equal to three. Since the conditions 
of equilibrium of the molecules in matter are independent of 
its state of aggregation, there should be continuity as we 
pass from the liquid to the gaseous state. The isothermal 
for a given temperature would therefore in certain cases be a 
curve of the well-known form a, d, c,d, e, shown in the figure ; 
the points b, c, d correspond to the three values of vat a given 
pressure. The points d and } correspond to the saturated 
vapour and liquid respectively at the same temperature. The 
pert of the curve between 6 and d is, however, not realizable 
in practice. Why this is the case will be discussed later. 


——-> PPESSURE 


_ VOLUME. 
At the critical point, denoted by E in the figure, the three 
values of v become equal to one another. ‘The equations 
giving the value of o when it has three equal positive roots 
in equation (7) are, according to the Theory of Equations, 
fear has + ae -Pag—0;4). -. = (a) 


df = 5 

Same a aaa ret 2. GB) 
af ; 

“To =e Pardo + Iagg=0;). 2.9. (e) 


Piul. Mag. S: 6. Vol. 20. No. 118. Océ. 1910. aX 


674 Dr. R. D. Kleeman on the Equation of Continuity 


where ay, ds, a3 are the coefticients of o’, o*, o° in equation (7). 
From equations (6) and (¢) we obtain 


700° + 28a,o° = 0, 


which, on making the necessary substitutions, reduces to 


at 


eT TY, 


Equations (a) and (/)) give the equation 


7a) + 4a,0' —3az = 0, 


which becomes 
ee Lxaey: KS e eis a en 


where K5 is the value of K, at the critical state and is the 
same for all liquids. We have already found a relation of 
this nature connecting the critical constants *. It is there- 
fore one of the conditions for the correcthess of the equation 
of state that it should lead to this equation or to equations 
from which it can be deduced. “a 

From equations (6) and (c) we also have wen 


300°— 12a,c? = 0, 


which, with the help of the results just obtained, reduces to 


This is the well-known law of Young and Thomas. The 
constant 2°1 is, however, too small, the mean value according 
to the facts is 3°7. 

The proposed equation of state thus leads to two known 
relations between the critical constants, but the numerical 
eonstants involved do not agree with the facts. It will 
therefore be necessary to introduce some modifications into 
the equation. These should first of all take into account that 
bis not a constant. Since we know nothing as to the exact 
nature of the variation of }, let us assume 


b = (n,—ngp). 


* Toe, cit. 


of the Liquid and Gaseous States of Matter. 675 


Equation (7) may then be written 


RT 
3 Ny 0/3 1 7/3 P & 51 =| P 
13/ oS 1 /: — — 2 — 6 — Ue 
Pp ny? a ny? zi wP A3ne e ¥ Asng 0 


Now, if we form the equations of condition for equal roots, 
it will be found that if we put 


where uw, and wv. are numerical constants, we obtain two 
equations of the form 


7/3 SS RT 
p= M, a) (> ,/m,)?, pP,=— Ea 
m mM, 


where M, and M, are functions of w, wz, and Ks. We have 
seen that we must arrive at equations of this form, and the 
numerical constants uw, and uw, must be so chosen that they 
agree with the facts. It is necessary first to obtain the 


value of K5.. We have seen that 


7/3 us 
P,=Up=A(£) (24/m,)?*, 
and Kg is therefore the value of A at the critical point. 
Without finding the exact nature of the variation of A with 
temperature, it will be seen from an inspection of the values 
of A of methyl formate in Table V.,t which have been calcu- 
lated up to the critical point, that the value of K3 is about 2800. 
If the pressure in the equation of state is expressed in 

atmospheres 
~ 26 7 
Ree See a INS orp e nat 

10° 

The mean value of M, can be deduced from the fifth column 
of Table IV.t, this giving M,;=(136°8)’. (Through an over- 
sight it was not mentioned in the paper that the values in 
this column are only relatively correct, the absolute values 
being obtained by dividing each value by 4°54.) We have 


* (a) pp. 794-795. 
T (a), p. 797. 
t (a) p. 788. 

2Y¥ 2 


676 Dr. R. D. Kleeman on the Equation of Continuity 


also M,=3'7. These two equations, substituting for Kg its 
numerical value, give u=="734 and up="L76. 

It is necessary next to discover a function which will 
express the variation of K, with temperature. This variation 
is small: thus in thé case of methyl formate the value of K, 
or A decreases from 4400 to 2865 when the temperature 
increases from 273 to 486. If the values of A given in 
Table V. quoted above are plotted against the temperature, 
it is at once apparent that they suffer from an accumulation 
of errors of data which affects them irregularly, but on the 
whole the values may be said to vary approximately linearly 
with the temperature. Bearing in mind that K, must be 
the same for all liquids at corresponding states, we may 
therefore write 

<i 
a 


It was found that we may put A=7222 and B=4422 ; at 


the critical state we have then K3=2800. The general 
equation of state is then 


999 9 T p ie AW, 


RTp 


= a 2 ey 
m{ 1-8 (734-1768) 
Pe P 


é 


But this equation, on account of its generality, cannot 
be expected to agree very well with the facts in all cases. 
A better agreement would be obtained by determining the 
numerical quantities separately for each liquid under con- 
sideration. These quantities would obviously, however, vary 
only slightly from one substance to another. 

It will be of interest now to compare the above equation 
of state with that given by van der Waals. Van der Waals’ 


equation is 
a RT 
(o+ 4) ~ m(v—by’ 
where a and 0 are constants which are supposed to be inde- 
pendent of temperature, &c., but which vary with the nature 
of the liquid. The term e or ap; corresponds to the intrinsic 
v 


pressure of the liquid. Now we have seen that this term 


313 
303 


of the Liquid and Gaseous States of Matter. 677 
must be equal to p,L’, or apj=L’p;, and therefore L’=ap,. 
Since the internal latent heat of evaporation L is given 
by L=L'—L” we have L=a(p,—p2), where a is constant. 
But this equation for the latent heat does not agree with the 
facts, and a is therefore not a constant. This is shown by 


Li 


Table I., which contains the values of at different 


P1— P2 
temperatures for a number of liquids; the values of this 
quantity or a, it will be seen, decrease considerably with 
increase of temperature. 


obtuin 
RI 
m2°66° 


But the constant 2°66 is too small, its value we have seen 
ought to be 3:7. Van der Waals’ equation thus satisfies 
only imperfectly two important conditions. The equation of 
state given in this paper satisfies these two conditions besides 
one other, and should therefore be in better agreement with 
the facts than van der Waals’. Moreover, it is more general 
in form and has a definite theoretical basis. 

The equation of state given in this paper was developed 


Pex. 


taking (5 B) or K in the expression for the attraction 


between two molecules as a function of the temperature only, 
in support of which we obtained some evidence. It will be 
of interest to develop an equation of state taking K a 
function of p and thus independent of the temperature. 
It should be observed that when the equation of state is 
applied to a liquid and its saturated vapour, it does not 


TaB_eE I. 
Ethyl oxide. | Carbon tetrachloride. | Methyl formate. 
| 
L L L L 
a TT. a ee . 4 are ape S . L . 
P,— p2 P,— P2 Pi—Pe2 Pi—Pe2 Pi— Pe Plo 
117-1 |393| 100-2 |/273| 29° |473| 25:3 |273| 1129 [443 | 890-5 
109'9 | 433 93-4 || 893 27°4 1513) 244 | 823} 1076 |483 73°4 
104-7 |460|. 88-7 |433| 264 1553) 22-4 |403) 956 4865) 681 
From the equations of condition for equal roots of v we 


678 Dr. R. D. Kleeman on the Equation of Continuity 


matter whether we consider K a function of the temperature 
or of the density, as the density is thus a function of the 
temperature. Since P,=Lp it will be most convenient to 
develop first a formula for the latent heat along the same 
lines. We have seen * that in general, according to the law 
of attraction between molecules, 


t= { (2)"-66(2.8)-G) (2) [a 


where 
d3 i ) and 3 (4 8) 
on oe 


have each the same values for all liquids at corresponding 
states. If K is to be independent of the temperature, 


¢; (=: 8) and ¢; (* ) 


must be functions of p, and p, respectively, or rather of 


Pi and P?, since they must have the same values at corre- 


c c 
sponding states. A fairly good agreement with the facts is 


obtained by writing for these functions 


u(2)" and u(e)” 


Cc 


respectively, where U is a numerical constant. The equa- 
tion for the latent heat then becomes 


> Vm)? 
L=U(p{—p3) ene ° 0) rete eh ae (9) 


This is the same equation as was obtained in a previous 
paper from surface-tension considerations +; it was there 
applied only to liquids considerably below their critical 
temperatures, and therefore written 


- (TV m4)? 


Io yae pl : 


hia 
The above equation is tested for a number of liquids over 
considerable ranges of temperature in Table IJ. The values 
of Lae 5 contained in the fifth and twelfth columns of the 


Page 
* (a) p. 746. 
+ Phil. Mag. Oct. 1909, pp. 499-5085. 


of the Liquid and Gaseous States of Matter. 


table are fairly constant for each liquid, as should be the 
case according to the equation. 
perfect in some cases than in others ; and it would therefore 
seem difficult to discover a simple formula involving p, and 
p, that would be in perfect agreement with the facts in the 


case of each liquid. 


Be 
; 


TasueE II. 
Ethyl oxide, C,H,,. M. wt. 74. Pentane, C,H,,. M. wt. 72. 
L 

oe py Piss 0292 K,.i|) 2. (1. Po L. (gna EK. K,. | 
273 | 7362 | -038270 | 86:16] 159°0 | 1486 | 1°73)| 273 | 6454 | -0,7756 | 85°85) 206°5 | 1914] 1-70 
293 |°7135 | 001870] 80°40) 158-0 | 1486 | 1:72)| 318 | 6062 | 00339 | 75:55) 205°6 | 1918) 1-69 
313 | 6894 | 003731) 75°36} 108-2 | 1489 | 1-72/| 333 | -5850 | -006024| 71°66) 209°3 | 1933] 1-71 
333 | °6658 | 006771) 70°79} 159-7 | 1497 | 1°73) 353 |-5624 | -01013 | 66°84) 211°3 | 1947) 1-71 
368 | 6402 | 01155 | 65°85) 160°7 | 1502 | 1°73)| 373 |-5378 | 01627 | 61:51) 212°7 | 1959} 1°71 
373 | 6105 | -01867 | 60°33) 161-9 | 1537 | 1°75|| 593 |-5107 | 02503 | 56-33) 215°9 | 1973) 1-73 
393 | 5764 | 02934 | 5491) 165°3 | 1530 | 1°75)| 413 | -4787 | 03861 | 49-08) 2156 | 1984] 1-72 
413 | -5385 | -04488 | 48°31) 167-8 | 1547 | 1°76)| 433 | -4394 | 05910 | 40°89) 215°6 | 1988} 1-71 
433 | 4947 | -06911 | 39°74| 165°6 | 1496 | 1°79)| 453 | -8867 | 09354 | 0°02) 213°4 | 1986} 1-70 
453 |:4268 |-1185 | 27-09} 160-0 | 1539 | 1-°68)| 463 | 3445 | -1269 | 21°11) 205°8 | 1957 | 1-66 
460 |-3663 |:1620 | 18-11} 167°7 | 1510 | 1-90/| 468 | 38065 | -1609 | 13°38) 196°8 | 1931] 1-61: 
466 | 33800 | -2012 | 12°03) 175°9 | 1464 469 |-2915 | 1745 | 10°55) 193-5 | 1918] 1°59 
Stannic chloride. SnCl,. M. wt. 260-8. Octane, C,H,,. M.wt. 114. 

373 | 20186) 005764] 28°68) 70°37 | 233-9) 1°73/| 273 | ‘7185 | 041942 | 84°71) 181-3 | 2415} 1°70 
393 | 1°9639| -00994 | 27°22) 70°50 | 235:3) 1°72)/ 393 | 6168 | 0033. | 64:03) 183°6 | 2347 | 1°79 
413 | 19073] 01616 | 25°67; 70°57 | 2386-9) 1°71) 413 | 5973 | 005464/ 61:13) 182°1 | 2360] 1°81 
433 | 18481} 02506 | 24:00} 70-28 | 237-9) 1°70) 433 | -5772 | 008584] 58:24) 181:0 | 2375] 1:84 
453 | 1°7873] (03759 | 22:08] 69°13 | 239-3) 1-66/| 453 | 5556 | 01314 | 54°43) 1783 | 2385} 1-85 
478 | 1°7224| ‘05450 | 20°23) 68:23 | 240-9) 1°63) 473 | 5317 | ‘01957 | 50-44 176-3 | 2390 | 1-86 
493 | 1°6488) (07728 | 18°28) 67:23 | 243-7] 1-59) 493 | 5053 | 02874 | 45°90} 1751 | 2419} 1-87 
513 | 15667) 1083 | 16°18} 66°19 | 242-2) 1°55)| 513 | 4732 | 04237 | 40°37) 171-4 | 2424! 1-88 
538 | 1°4747| 1520 | 13°88) 64°59 | 238-7) 1°52|| 533 | 43864 | 06223 | 34-25) 168°5 | 2418] 1-90 
553 | 1:3628) 2160 | 11:27) 60°80 | 221-2) 1-45)| 553 | 3818 | 09833 | 24-67; 164:2 | 2395 | 1-89 
Ethyl propionate, C,H,,O,. M.wt. 102. Sulphur dioxide, SO,. M. wt. 64. 
273 | 9142 | -0,4850) 91:03} 1089 1396 | 1-74] 263 | 1-460 | -002964] 86:56] 44:19 | 359-1| 1-73 
383 | °7823 | 004739) 70°17} 114°7 | 1391 | 1°84); 283] 1-410 | -0068 | 78-46) 39°47 | 329-8} 1-68 
403 | °7548 | 008000) 65°58; 115:°0 | 1896 | 1°84) 803 | 1353 |-0134 | 73°32) 40:05 | 333-3] 1-69 
433 | 7115 |-01615 |58-09} 114-8 | 1406 | 1°82)/ 323 | 1-296 |-0250 | 65:12) 38°76 | 329-6] 1°65 
453 | 6795 | -02469 |53-10) 115-0 | 1413 | 1°82) 343 | 1:233 | -0396 | 62:75) 41:29 | 337-0] 1:72 
473 | 6443 | 03676 |48:24| 116°5 | 1422 | 1:83) 363] 1:158 |-0608 | 58-38] 43°66 | 347-4) 1-76 
493 | 6027 | 05435 | 42°63) 118°3 | 1480) 1°85/| 383 | 1-070 |-0995 | 48-48) 42°67 | 348-1) 1-74 
5138] -5501 | 08230 | 35°17) 118-9 | 1431|1-86) 403] -960 |-191 | 39-46) 44:06 | 348-9) 1:82 
533 | 4744 |-13837 | 24:19) 116°3 | 1415/ 1-84) 418] -827 | -254 28°57) 46:19 | 3465) 1:87 
541 | 4227 |-1751 | 16°85) 113°8 | 1401 |1°81)| 428] +650 | -420 11:26) 45°72 | 329-9) 1°94 


679 


The constaney is more 
yi 


OU ESS EE eas RPE A RT ESN PRT EA Se A a a Sd a LURE 


680 Dr. R. D. Kleeman on the Equation of Continuity 
Table II. (continued). 


Benzene; C,H,. M. wt. 78. | Heptane, C,H,,- M. wt. 100. 
L dil L 
5 ed pi: 02: bs. Eee Bo | ea aie se Pa: L. 9, ae B. | Ky. 
273 | 9041 |-0,1215) 100-1] 123-6 | 1305 | 1-63) 273|-7005 | -0,6725| 84-44] 172-1 | 2236) 1-69 
398 | °8145 | -0U2722) 85°62) 129-0 | 13817 | 1°68)) 353 | 6311 | 001996} 72°74) 186-0 | 2216} 1°81 
373 | °7927 | -004690) 81°98) 133°6 | 1322] 1°69) 373 | 6124 | 003584) 68 95] 183°8 | 2220} 1°81 
393 | °7692 | 007634) 78:12} 132°0 | 1330 | 1-70)) 393 | 5926 | -006068) 64:67) 1842 | 2228 | 1-81 
413) -7440 |-01174 | 74-09) 133-9 | 1845 | 1°71] 413 | 5711 | 009775] 60:07] 184°2 | 2238/ 1-80 
433 | °7185 | ‘01734 | 69°74} 135-1 | 13856] 1-71) 433 | 5481 | -01508 | 55°69) 185-4 | 2250) 1°81 
453 | 6906 | 02487 | 65°12} 136°6 | 13872} 1°71)| 453 | 5232 | 02242 | 51°62) 188°6 | 2271 | 1:82 
473 | 6605 | 03546 | 59:95) 137°5 | 1877 | 1°71) 473 | 4952 | 03304 | 46°68} 190-1 | 2269] 1°84 
493 | 6255 |-05015 | 53°84) 138°8 | 1393] 1-71|| 493 | -4616 | 04892 | 40°57] 192-6 | 2281 | 1-85 
513 | 5851 |°07138 | 46°63} 148-2 | 1390} 1°71)| 513) -4177 | 07446 | 32:60} 193-0 | 2279 | 1:86 
533 | 5328 |-1038 | 37-49} 137°3 | 1387 | 1-70)! 5383 | 3457 |-1287 | 19°02} 184-9 | 2293 | 182 
553 | "4514 | "1660 | 23°45) . 135°6 1892 | 1°67) 589 | 2907 | 1778 9°25} 175°2 | 2177 | 146 
Jet ee 
N. Hexane, C,H,,. M. wt. 86. | Carbon tetrachloride, CCl,. M. wt, 154. 
275 | ‘6770 | 0,2268 84:68} 184°8 | 2069} 1°68) 273 | 1:6327| 0,2984| 48°35) 18:12 | 486-5) 1-40 
343 | 6122 | 00337 |'71-81} 191°6 | 2067 | 1:75)| $73 | 14343) 01026 | 39°68) 19°31 | 889-1) 1-67 
363 | 5918 0585 | 6791} 193°9 | 2078} 1-76)| 393 | 1°3902) 01634 | 37-63) 1947 | 392-6) 1:67 
383 | °5703 | 00952 | 64-01} 1968 | 2094 | 1-77|| 413 | 13450) 02481 | 35-56) 19:65 | 3895-9) 1°67 
403 | 5467 | -01502 | 59-10} 197-7 | 2104] 1°77)| 438 | 1:2982) 03650 | 83:28} 19°73 | 398 4) 1-67 
423 | 5207 | 02299 | 53-61] 197-7 | 2114/ 1-76)| 453 | 12470} 05249 | 30°83) 2000 | 400-2) 1-66 
| 443 | 4913 | 03472 | 47-41) 197-3 | 2122) 1°75)| 473 | 11888) 07418 | 28:22) 20-01 | 404-2) 167 
463 | °4570 | :05155 |40°91} 1985 | 2128 | 1°76} 493 | 11227) 1040 =| 25°35) 20°28 | 407-1; 1°68 
483 | 4124 | 07900 | 32-20} 196-6 | 2116 | 1°75|| 513 | 10444! 1464 | 21-91) 20:54 410°3 1 69 
| 499 | 8557 |-1208 |21-33) 190-2 | 2095 |1-71/|533| -9409) 2146 | 17-15] 20-44 |4078 169 
506 3040 |-1658 | 1176} 1809 | 2050 1:66] 553) “7634 3597 | 890] 1963 | 3986 166 
Iodobenzene, C,H,I. M. wt. 203-9 Bromobenzene, C,H,Br. M. wt. 157. 
403 | 18149, 0,195 | 58°75, 16-31 | 464-9] 1-57| 403 | 1-4815| 0.4702! 64 84| 29-51 | 620-4] 1-64 
503 | 17079, (4400 | 50 06} 17:19 | 459°7| 1°66 | 583 | 1-2994) 005255) 49-62) 29°40 | 613-9) 1°65 
563 | 1-5627 -006020 40:97; 16:78 | 457-6) 1:64 553 | 11-2697) 0O&071! 48°51) 80-07 | 617-0] 1:67 
583 | 1°5316) C08&89| 40-72) 17°35 | 460°3) 1°68 | 573 | 1-2385) 01205 | 46°85} 30°53 | 620-1] 1°69 
603 | 1°4941 -01296 | 39°72) 17°79 | 464-6) 1°71] 593 | 1-2037| 01750 | 44:93) 30°99 | 625-2) 1°70 
| 623 | 1°4581) 01849 | 38°59) 18:15 | 466-8) 1-74| 613 | 1:1689| 02482 | 42°80) 31°30 | 627-7) 1°72 
643 | 14172 -02605 | 37°29) 18°58 | 469°6 1°73,| 683 | 1-1310) 03427 | 40°82) 31-91 | 622°7| 1°73 
Di-isobuty], C,H,,. M. wt. 114. Ethyl acetate, C,H,O,. M. wt. 38 
73| °7102 | 044762 | 76°24) 151:1 | 2259] 1°67 | 273 | °9244 | 031255 | 94-45) 110°5 | 1235} 1°72 
373 | -6236 | (002967| 63°05| 162-1 | 2227 | 1°82 | 363 | -8112 | 004673! 76°47; 119-0 | 1234 | 1°81 
393 | -6046 | 005236) 57-63) 157-7 | 2220} 1:78 | 383| *7831 | -008U00| 72:19} 117-7 | 1243 | 1°82 
413 | -5841 | 008532) 53°65) 157-2 | 2222) 1°77 | 403} 7533 | 01812 | 66 93) 1180 | 1248 | 1°82) 
433 | -5620| 01319 | 50:11) 158-4 | 2236 | 1°77'| 423] -7210| 02062 | 6166) 1187 | 1256] 1:82 
4538 | 53883 | -01¢57 | 46-90; 161-9 | 2250| 1:80] 443] -6848 | 038165 | 55-71) 118-8 | 1261 | 1:82) 
473 5117 | 02874 | 42-91) 163°9 | 2260) 1°81\|463| °6441 | 04751 | 49-48) 119°9 | 1270) 1:82 
493 | -4810| 04202 | 38-32) 166-9 | 2275 | 1:83} 483) -5944 | 07128 | 4208} 120-8 | 1277| 1-82 
51% | 4434 | 06223 | 32-59; 169-0 | 2270} 1-86 | 503) -5281/-1131 | 31-32) 117-7 | 1265 | 1-80 
533! *3912 | 09699 | 24°31) 169-2 | 22477 | 188/518) 4401 | 1802 | 18°00) 111°7 | 1283 / 1°75) 
547 | -8187 | -1572 12°39} 161-7 | 2183 | 1-85] 522) -3893|-2288 |10°41} 94°3 | 1104 1:66 


t 


— eS 


a 


SO 


a 


Fluor-benzene, C,H,F]. M.wt 96:1. 


if, 


273 
353 
373 
393 
413 
| 433 
453 
473 
493 
513 
533 
503 


Py: 


1°0465 
9496 
9233 
"8955 
‘8665 
8363 
8037 
‘7671 
"7265 
6789 
6163 
5138 


of the Liguid and Gaseous States of Matter. 


Pos 


031179 
‘002885 
‘005040 


008333 


01321 
01992 
02911 
04184 
‘05907 
‘08403 
1226 

‘2034 


Toes 


81°74 
72 96 
69°71 
65°68 
60:97 
56°86 
52°69 
48:18 
43°57 
38-00 
80°75 
18:00 


Table I. (continued). 


681 


Carbon dioxide, CO,. _M. wt. 44. 


K4. 


Hexamethylene, 


273 
363 
383 
403 
423 
443 
463 
483 
503 
523 
543 
5d2 


7967 
7106 
6898 
“5680 
6448 
6200 
5926 
5626 
‘5271 
4820 
4125 
3393 


Coles. 


M. 


wt. 841. 


0,1374 
003759 


‘006289 


‘019000 
01508 
‘02183 
03140 
04437 
‘06250 
‘09058 
"1433 

°2105 


Methyl formate, 


273 |1:0032 


303 
(323 
043 
363 
383 
403 
423 
443 
463 


483 


9598 
9294 
"8968 
8634 
"8264 
"7860 
7403 
"6844 
‘6148 
"4857 


0,6821 
QU2225 
004396 
007968 
‘1352 
‘02153 
03344 
05063 
07634 
‘1178 
2188 


64:08 


89°77 
(thas! 
72°75 
67°52 
63:08 
59°12 
53°98 
48°80 
42:97 
30°15 
22°85 
10:07 


C,H,0, 


113°2 
107°5 
99°51 
92°16 
85°10 
79°21 
71°95 
544] 
41-95 
19°58 


152°7 
1420 


112°6 
ties 
115:2 
114-7, 
114-1 
116-0 
116°6 
117°4 
117°6 
115°2 
104-2 


1618 
1636 
1642 
1649 
1660 
1674 
1683 
1699 
1700 
1690 
1670 
1614 


Pr a 
DPDAAMWAARABWAIAIRD 


NO OH WOO WO mS BP 


M. wt. 60. 


8600 
866°6 
869°4 
8748 
879:0 
888"4 
895'2 
9$03°6 
912°0 
912°4 
890°0 


fl pe ee el pet ee Rt et 
SST TUNA 
me ODOR rte be ATL 


Ate 


243 
268 
283 
298 


px 


1-045 
"956 
846 


‘705 


0725 
135 
"253 


iby, 
65°34 
54:45 
40°56 
22:07 


| 
Ku. 
1-74 
1-78 
1-73 
1-65 


Chlorobenzene, C,H,Cl. 


273 
413 
435 
453 
473 
493 
5138 
533 


11278 
9723 
"9480 
"9224 
"8955 
"8672 
"8356 
‘8016 


‘041689 
‘004316 
‘006761 
01020 
01500 
02145 
03000 
‘0417 


82:90 
66°48 
64°12 
61°46 
58°31 
59°29 
52°43 
49:09 


Chlorvform, CHCl,. 


M. 


65°19 
70°33 
71°36 
72:24 
(Pav 

73°52 
75°08 
76°42 


M. wt. 


1036 
1028 
1035 
1041 
1048 
10538 
1062 
1090 


119 


Whe lame 


tel att at tt 
AG eS ge BON ON Mor) 
ORNrRr CON 


“Eye 


273 
293 
313 
333 


279 | 


293 
313 
333 


1:5264) -034027 
1-4885} 001042 


62°45, 
60°14! 


26:93 
27-11 


1:4503} 002248) 5'7°87| 27°52 
14108] 004356) 55°60) 27-94 


415°5 1:35 
416-8) 1-70 


418°2 
419-9 


1°72 
174 


Acetone, C,H,O. M. wt. 58. 


8186 
‘7960 
7731 
7497 


"032339 | 131-8 
"035688 | 127-2 
001215) 121-4 
002372) Vi72 


196°6 
200°7 
203°1 
208°5 


Ethyl formate, C,H,O,. M. 


273 
333 
353 
373 


393 
413 
433 

453 


473 


493 
| 303 
507 


9480 
‘8689 
8409 
"8112 
‘7796 
"7448 
"7058 
“6610 
“6066 
"D290 
"4635 


‘4117 


033152 
°YU3356 
006061 
‘01031 


‘01656 
02558 
‘03876 


‘05747 


‘08621 
"1379 
"1890 
"23538 


99°50 
87°56 
82°54 
76°83 
71°33 
65°63 
58°80 
51°64 


42:54 
29-59 
19°51 
11-89 


110°8 
116-0 
116°7 
1168 
Togs 
118-2 
118-0 
1181 
118-0 
113-4 
100°1 
104°3 


1437 
1455 
1469 
1481 


aaa aqsaaa a4 
RIAA AWS SO 


eet et et tO 


STN Op te 


Oc: 


682 Dr. R. D. Kleeman on the Equation of Continuity 

The third column of Table III. contains the mean value of 
= of each liquid contained in Table II. According to 
equation (9) this quantity is equal to 


U ile 
7/3 ,2/3 (> Md my)”. 
Tae 


TABLE ITT. 


; He L 2073(2%m)2 | _ | 258:8(2 ¥m,)? 
Name of Liquid. Po. p.?—p,2 527k fee sh mi %p2/8 ; 

Chlorobenzene ......... 3654 | 71:49 66°83 1049 938-7 
PS) (odbc 3 Un. oneh 9393 | 209°4 DAW tar § 1951 1960 
ELCDUMO.,., <ivonanysacres ‘2341 | 185°0 1586 2246 1982 
Stannic chloride ... .. ‘7419 6779 7009 237°0 228'2 
RIE 5), de bic dds 9344 | 193°3 1787 2094 1923 
Ethyl oxide ............ 2604 | 163°3 171-3 1507 1582 
PICTIAGNO nig vce dncs aks 08 3045 | 1384°3 126°3 1357 1228 
Todo-benzene............ ‘5814 | 17°45 16°67 463°4 424°4 
Hexamethylene ...... ‘2735 | 151°8 191‘8 1661 2014 
Carbon dioxide......... "464 BUST 66-43 322°7 365'1 
RAO ee usb vs we 2327 | 176°2 165°5 2393 2358 
Etbyl propionate ...... 286 | 115°3 1225 1410 1559 
Carbon tetrachloride. |°5576 | 19°74 17°95 403'7 3449 
Ethyl acetate ......... -2993 | 115°3 119°8 1239 1308 
AMSODUEYL 03.30.00 2366 | 161°7 1791 2247 2552 
Fluor-benzene ......... *3541 81°44 82°30 1001 987-4 
Bromobenzene ......... *4853 | 30°53 30°39 621:0 595°7 
Methyl formate ...... 3489 | 115°4 1063 893-9 79672 
Ethyl formate ......... | ‘815 | 1139 115°8 1092 1069 


The values of the latter quantity were calculated, and are 
given in the fourth column of the table, U being put 
equal to 2073. The value of U is the mean of the values 
obtained by equation (9) for the liquids. The agreement 
between the two sets of valuesis fairly good, and equation (9) 
may therefore be used to obtain the approximate internal 
Jatent heat of evaporation of a liquid at any given 
temperature. 


The values of ites for a large number of liquids have a 

Ly phe 
tendency to increase slightly with the temperature. It was 
therefore thought desirable to test the expression using a 
different power of p than 2, preferably less than 2. The 


L 
values of oi pBB have therefore been calculated for carbon 
1 2 


+ 


of the Liquid and Gaseous States of Matter. 683 
tetrachloride and ethyl propionate, and are contained together 


te 5 in Table TV. It will be seen that 


1 2 


with the values of 


TABLE LV. 


Carbon tetrachloride. 


| 
| 


a 


tp -/18°12 19°31 Loa 19°65 19°73 |20°00 |20°01 |20°28 |20°54 |20°4 119-63 
1 2 | | 
eee 21°34 [21°77 !21°74 |21°69 |21°53 |21°31 |21°3) |21°50 [21-23 |20°77 |19°54 


Ethy1 propionate. 


L 
are 108°9 114°7 1150 114°8 cate 118°3 |118°9 |116°8 |113°8 
1 2 ] i 
—————$ —_ - | —_ | |_| [= Seal —— 
i? 
02/3—,5/3 "114-1 105°6 |104°8 |102°4 |101°1 ee 100°2| 99°4 95-4 | 89:9 
1 2 


t 


1 


the values of the former expression are more constant than 
those of the latter for carbon tetrachloride, but the opposite 
is the case with ethyl propionate. The equation for the latent 
heat according to the former expression is 


La Ute —p7} ema 


7/38 1/3” 
mi! p,!? 


It appears therefore that in some cases this equation will be 

in better agreement with the facts than equation (9), while 

in other cases equation (9) will be in better agreement. 
Since P,=L'p, we may now write 


2x 2073 2 (S./my 
P, = 42 x 2073 a ( J/m,)” 


and the equation of state becomes 


Pile ap ee RTp 
Tee > ee 
p+ 87066 japan ( vm) Tet kal: (10) 


taking 6 first of all as constant. From the equations 


* 


684 Dr. R. D. Kleeman on the Equation of Continuity 


expressing the conditions for equal roots we have 


87066 i ae pe 
Se ae @ 2m)? and P.= m5 


These two equations, we see, are of the required form but 
the numerical constants have not the proper value. We will 
therefore, as before, assume b=(n—n pe). From the equa- 
tions of condition for equal roots we then have that if we 
put 


we obtain two equations of the above form connecting the 
critical constants; and if we further put w,="06 and w.=*602 
the numerical constants in the equations will have the 
proper value. The above equation of state and the one 
given previously will be further discussed later in this 
paper and in subsequent papers. It may be noticed here 


that 
: (-602— os 06), 
Pe 


c 


the expression obtained for 6, is of the same form as the 
value obtained previously, viz. : 


~(-734- P-176). 
P. Pe 


The reason that the part bed of the curve in the figure is 
not realized in practice does not seem to have yet been 
made quite clear. It is intimately connected with the 
property of a liquid and vapour to be able to exist in equi- 
librinm side by side ; in fact, it appears that in all cases 
where two portions of matter of different densities can be in 
equilibrium in contact with one another, the states corre- 
sponding to the intermediate densities cannot be realized 
in practice. According to the equation of condition of a 
molecule in the liquid or gaseous state, it follows that if 
each molecule is in the same condition, two portions of matter 
of different densities cannot exist in equilibrium in contact 
with one another. The matter of less density would con- 
dense upon that of the greater density. Therefore, since a 
vapour can exist in contact with the liquid, the molecules 
must differ from one another in some way. The explana- 
tion is that the molecules differ from one another in their 


of the Liquid and Gaseous States of Matter. 685 


velocity of translation ; further, the velocity of each molecule 
is continually changing, the distribution of velocities among 
the molecules at any instant being given by Maxwell’s law 
or some law similar to it. A certain number of molecules 
will therefore each second obtain sufficient kinetic energy to 
be able to get away from the attraction of the molecules of 
the liquid, equilibrium being produced when the number 
shot out of the liquid is equal to the number returning from 
the vapour. When the volume of the vapour is decreased hy 
compression its density must remain the same, for the 
number of molecules shot from the surface of the liquid per 
cm.” is unaltered by the process, and therefore the number 
coming from the vapour must also remain unaltered, and a 
portion of the vapour must therefore condense to keep the 
density constant. 

Now, suppose a cylinder which has one end closed and in 
which a piston works contains some saturated vapour of a 
liquid, and suppose we endeavour by moving the piston to 
pass the vapour through those changes usually not realized 
in practice. From a consideration of the law of attraction 
of one molecule on another and the average density of rigid 
materials, it follows that the attraction of the material of the 
cylinder on the vapour will be very nearly equal to that of 
the liquid corresponding to the vapour. The surface of the 
cylinder will therefore be covered with a thin layer of vapour 
which will be nearly as dense as the liquid. A small 
increase of pressure only will therefore in general be 
required to increase the density of this thin film, so that it 
is equal to that of the liquid. When that stage is reached, 
condensation of the vapour upon this liquid film takes place, 
and its pressure then decreases till it is equal to that of the 
ordinary pressure of the vapour in contact with the liquid. 
The pressure remains constant on the volume of the vapour 
being further decreased, and the vapour thus does not pass 
through the relations between pressare and volume indicated 
by theory. It appears, therefore, that if it were possible to 
construct the cylinder out of material which has no molecular 
attraction, the vapour could be passed without difficulty 
through all the isothermal changes indicated by theory. 

From the equations of state given in this paper we can 
deduce some relations of interest and importance. Referring 
to the figure, it follows from thermodynamics* that the 
amount of external work done in passing along the isothermal 
from b to d is equal to the work done in passing from b to d 


* Winckelmann, Handbuch der Physik, Wérme, p. 654, second 
edition. 


656 Dr. R. D. Kleeman on the Equation of Continuity 


along the straight line. This well known condition is 
expressed by the equation 


( p .dv=p(v;—22). 


Substituting for p on the left-hand side of the equation 

from equation (10) and integrating, we obtain an equation 

which may be written 
Be oh ci 0h. (ER 
5 (Pi—p3)— log ma +h=p(v.—0), 

where h is a positive quantity much smaller in magnitude 

than the term which precedes it, and 


87066 a 
P= agi (SVM) a, 
Now P(p{—p3) is the internal heat of evaporation per gram 
of substance and p(v,—v,) the external work done during 
evaporation, and the latter quantity is therefore much 
smaller than the former. The quantity (p(ve—1)—h) is 
therefore probably very small in comparison with the first 
term in the equation, and the first two terms therefore of 
the same magnitude. We may therefore suppose 


aP 
5 (pi— pz) + p(r—r1) -—h=0 


in the equation, where z is a very small fraction which is 
taken as constant, and we therefore have 


B(pj—p}) =T log”, ‘hry oe 


where E is a constant. This equation was tested in the 
case of a number of substances over considerable ranges of 
temperature. The result is given in the sixth and thirteenth 
columns of Table II., which contain the values of E calcu- 
lated by means of this equation. It will be seen that E is 
remarkably constant for each substance. This equation thus 
gives very accurately the relation between p,, p2, and T, 
for different temperatures of a jiquid. 


The value of E is proportional to -_ and therefore pro- 


ror aN 
portional to (S/n)? The fifth column of Table III. 


4/32/83 
Te (P, 


iia mice ie 


of the Liquid and Gaseous States of Matter. 687 


contains the mean values of KE of each liquid in Table IL., 
and the sixth column contains the values of 


2588 (24/ mz)? 
oO eee 
Pe 


Ii will be seen that the two sets of values agree approxi- 
mately with one another. The constant 258°8 was obtained 


by dividing the values of E in Table II. by the corre- 


2 

sponding values of aa: and taking the mean of the 
values obtained. ke 

In the equation expressing the equality of the work done 
in passing from 6 to d in the figure either along the straight 
or curved part, if we substitute for p from equation (6) and 
integrate we obtain, assuming that the sum of certain terms 
in the equation is zero, in a similar wav as before, that 


Sot? py") =T log He ab dais ive (BBD) 


where 


erie ewe 
Ts 


and dis a numerical constant. ‘This equation should be in 
approximate agreement with the facts, since it is simply 
the left-hand side of equation (11) expressed in a different 
way. The-left hand sides of both equations (11) and (12) are, 
according to equations (3) and (9), it will be observed, 
equal to Lm multiplied by a numerical constant. Ky, we have 
seen is a function of the temperature only and is approxi- 


T : 
mately given by (7222-4422 T)” Its exact form will be 


investigated in a subsequent paper, . 

At the critical point the value of E given by equation (11) 
is an indeterminate fraction, and it will therefore be of 
interest to determine its limiting value at that point. Let 
p2=2p; and we have 


2a? 


is gS, am 
log, :) 
ygpa Te (3) ieee oe, 
= z=!) p22 


changing in the beginning the logarithm from the base 10 


683 Dr. R. D. Kleeman on the Equation of Continuity 


to the base e. Now, E must have the same value at the 
critical temperature as at lower temperatures. Substituting 
for E its value 


in the above equation, we have 


4/3 
T,= 11896 (82) (S/n? 


ML 


This relation between T., p,, m, and my, has already been 
obtained by the writer in a previous investigation*. The 
numerical coefficient in the equation obtained in the above 
way is also of the proper magnitude. Thus the value of the 
coefficient calculated by means of the equation using the 
critical data of ether is 1127, which is approximately the 
same as that given above. Equation (12) also leads to the 
above equation. 

We have seen that each of the left-hand sides of equations 
(11) and 12 is equal to Lm multiplied by a numerical 
constant. This gives another formula for the internal heat 
of evaporation which may be written 


_K BT, 
h=K, Csi, 


where Ky isa numerical constant. If w in the equation 


(13) 


er be 
* (p2—p2) + p(v.—01) —h=0 


is zero, then it follows from the equation from which equa- 
tion (11) is derived that Ky, is equal to 2. The actual value 
of K, was found to be equal to about 1°75. ‘This is shown 
by the seventh and fourteenth columns of Table I]., which 
contain values of Ky, calculated by means of equation (13). 
It will be seen that K, is not quite independent of the tem- 
perature, it usually increases slightly with the temperature 
till near the critical point and then decreases again. The 
constancy of Ky, is further tested in Table V. for several 


* Phil. Mag. Dec. 1909, p. 906; (a) pp. 783-787. 
g P 


of the Liquid and Gaseous States of Matter. 689 


TaBLE V. 
ae Propyl Methyl Propyl Methyl 
Di-isopropyl. | Isopentane. formate. | propionate. acetate, butyrate. 


‘Fe ie is WiC ge Sa Aaa fo K4. Per eles |? K4. 


mie. | 1°68 | 273 70) 2i8 168) 275 | bil | 27s 172) 273 | 173 
333 | 1°74 | 293 |1:67| 363 /|1:78| 353 | 1°80 | 373. |1°84| 383 | 1:78 
353 | 1°74 | 313 |1°67| 383 |1°77| 373 | 1°80 | 393 |1°85) 403 | 1°80 
873 | 1:70 | 333 |1°68| 403 |1°78) 393 | 1°80 | 413 |1°86) 423 | 1°81 
393 | 1°73 | 353 |1:69| 423 |1-77| 413 | 1:80 | 433 |1°86| 443 | 1°81 
413 | 1°73 | 373 |1:69| 443 |1°76| 433 | 1°81 | 453 |1°86} 463 | 1-83 
433 | 1°72 | 393 |1:69| 463 |1°78| 453 | 1°81 | 473 |1°86] 483 | 1°84 
453 | 1°73 | 413 |1°69| 483 |1°78| 473 | 1°82 | 493 |1°86} 503 | 1°85 
473 | 1°73 | 483 |1°69| 503 |1°80| 493 | 1°82 | 513 |1°86| 523 | 1:86 
489 | 1°70 | 449 |1°66| 523 |1°78| 513 | 1:79 | 5383 |1°84|) 543 | 1:83 
498 | 1:64 | 458 |1'61/ 533 [1°79] 528 | 169 | 546 |1°76| 553 | 1°72 


liquids not mentioned in Table II. The values of py, po, and 
L, used in the calculations, are not given in the table: they 
can be obtained from tables given by Mills, which were quoted 
previously in this paper. The calculations have been carried 
out up to a few degrees below the critical temperature. 
It will be seen by inspection that the mean value of K, 
for each liquid depends slightly on the nature of the 
liquid. 

It should be mentioned here that Jager, Voigt, and 
Dieterici* have arrived at equations for the internal latent 
heat which resemble more or less equation (13). These 
equations were obtained from considerations of the kinetic 
equilibrium between the molecules shot out of the liquid 
into the surrounding vapour and the molecules returning 
from the vapour to the liquid. Dieterici, in the paper men- 
tioned, by making certain assumptions to simplify the result 
arrives at the same equation as the above. The application 
of the equation to a few liquids showed that K, is equal to 
about 1:7. A much more comprehensive test of the equation 
is given in this paper. 

Cambridge, 

June 2, 1910. 


* Ann, der Phys, xxv. p. 569 (1908). 


Phil. Mag. 8. 6. Vol. 20. No. 118. Océ. 1910. 2Z 


ees | ee | | ee | | | 6 | | 


iit se Wich tris Atk 


[ 690 ] 


LXXIV. The Scattering of Waves by a Cone. By Professor 
H. 8. Carstaw, The University of Sydney, N.S.W.* 


al view of the interest at present taken in the question of 

the scattering of waves by a sphere, the corresponding 
problems for a cone may have some slight value. Recently 
I have obtained the expression in series which gives the 
solution for the cone, but I have not yet been able to reduce 
my results to a form suitable for numerical discussion. The 
method which I follow is similar to that of a former paper 
on Diffraction +, and is suggested by Dougall’s work on 
Potential {. The proof is hardly suitable for these pages, 
and I confine myself for the present to a statement of one of 
the results obtained. 

The vertex of the cone is taken as the origin. Its surface 
is given by 0 = @), and its axis by 0 = =. 

We start with a source at the point on the axis produced, 
at a distance 7’ from the vertex. This is the point (7, 0, 0) 
in spherical coordinates. 

The disturbance in the infinite medium due to this source 
is defined by 
o7ikR 

R.” 


where R? = 24+ 77—2rr' cos @. 


Up = 


This can be written 


T 


i] 
9e4+ @ nim Aan 
a/ VY 0 2 2 


for r<r’, with the usual notation for the Bessel’s Functions §. 

On replacing this series by an equivalent Contour Integral, 
and associating with it the solution required by the surface 
condition u = 0 at 86 = 4, we obtain the following result :— 


2 fc 5 ‘ lings oi ge 

us ga ls ee eae K 1 (er) Ja ej 
A ; | sin nr ad P.,(Ho) 

for r<r’, thé summation being for the values of n > — 4 which 

make P, (™,) vanish. 


* Communicated by the Author. 

+ Phil. Mag. (6) vol. v. (1903). 

{ Proc. Edinburgh Math. Soc. vol. xviii. (1900). 
§ Cf. Macdonald, ‘ Electric Waves,’ p. 91. 


Number of « Particles emitted by Uranium. 691 
hea 
‘ do 
case, and allows us to put this result in the form 

4 oe alse P,, 
ea K 2 (ixr') J # (Kr) Nyneleded 


an ae d al ni 
vn (1— a’) 5, Pa (oo) Pao) 


for r<7’, the values of n being as above. 
The symbols 7, 7’ have to be interchanged when r>7’, and 
if the source were at (7’, 6’, g') instead of at (7’, 0,0), a 
corresponding, but more complicated, result would hold, 
The problems in conduction of heat analogous to these lend 
themselves to the same treatment. 


Sydney, June 1910. 


A relation between P, (—o) and =— Pa (p,) exists in this 


> 


LXXV. The Number of a Particles emitted by Uranium and 
Thorium and by Uranium Minerals. By Hans GEIGER, 
Ph.D., and Professor E. RurHErRForD, /’.R2.S.* 


|‘ previous papers we have shown that the number of 
i. «a particles emitted per second from radioactive materials 
can be counted either by the electrical or scintillation method. 
It has been shown that one gram of radium itself, and each 
of the three « ray products in equilibrium with it, emits 
3°4x10!° a particles per second. Since Rutherford and 
Boltwood ¢ have shown that in an old unaltered mineral 
there is 3°4x 1077 gram of radium per gram of uranium, it 
is possible to deduce the number of « particles emitted per 
second from one gram of uranium and also from a mineral 
containing one gram of uranium. In this calculation it is 
supposed that uranium is the ultimate parent of radium, and. 
that the mineral is in radioactive equilibrium. If a uranium 
atom, like a radium atom, emits one « particle in its trans- 
formation, the number of @ particles emitted per second per 
gram of uranium should be 3:4 x 10 x 3-4 x 107’, or 11,600. 
We shall for convenience call this number N. 

As a result of a very careful analysis of the radioactive 
constituents of uranium minerals, Boltwood { has shown that 
the total activity of uranium, measured by the electric 
method, is about twice as great as would be expected if 


* Communicated by the Authors. 
+ Amer. Journ. Sci. vol. xxii. p. 2 (1906); also Boltwood, Amer. 
Journ. Sci. vol. xxv. p. 296 (1908). 
t Boltwood, Amer. Journ. Sci. vol. xxv. p. 270 (1908). 
2 dae 


ee 


692 Dr. Geiger and Prof. Rutherford on the Number of 


uranium emits one a particle for one from the radium itself 
in equilibrium with it. This suggests that the uranium 
atom in its transformation emits at least two & particles. In 
the present state of our knowledge it is not certain whether 
this can, be ascribed to the existence of an additional ray 
product which is always separated with the uranium, or to 
the expulsion of two or more & particles in the transforma- 
tion of the uranium atom. 

Supposing, for the purpose of calculation, that the uranium 
in a mineral emits two « particles for one from each of the 
subsequent six a ray products, viz. ionium, radium emana- 
tion, radium A, radium ©, radium F (polonium), the number 
of « particles emitted per second per gram of uranium in a 
mineral is 8 N, or four times the number emitted by ordinary 
purified uranium. In this calculation no account has been 
taken of the actinium which occurs in all uranium minerals, 
and which Boltwood has shown stands in a genetic relation 
with uranium. However, Boltwood (loc. cit.) has found 
that the actinium and its four e ray products contributes an 
activity to the mineral equal to only ‘21 of that of the 
uranium. The relative number of « particles is still smaller, 
for the a particles from actinium have an average range of 
about 5°7 cms. of air, while the a rays of uranium, according 
to Bragg, have a range of 3°5 cms. Taking as a first 
approximation that the ionization due to an « particle is 
proportional to its range, the number of a particles emitted 
by the actinium in a mineral should be about °17 of that 
from uranium. ‘The total number of « particles emitted by 
a mineral containing one gram of uranium should con- 
sequently be 2°34 N+6N = 834N. Since N by calculation 
is 11,600, the total number of @ particles emitted per second 
from a mineral containing one gram of uranium should be 
9°67 x 10+, and the number per second from one gram of 
ordinary purified uranium should be 2°32 x 10*. 

It was the object of the present experiments * to deter- 
mine the number of 2 particles experimentally, and to test 
the agreement with the calculated number. 


* The experiments described later were, for the most part, completed 
more than a year ago. Recently, J. N. Brown (Proc. Roy. Soc. vol. A. 
lxxxiv. p. 151, 1910) has counted the scintillations from a uranium 
mineral and found a value per gram of uranium of 7°36 x 10*, which 


is somewhat smaller than our experimental value given later, viz. 
9°6 x 10* 


a Particles emitted by Uranium and Thorium. 693 


Arrangement of Experiment. 


The scintillation method was adopted in order to count 
the number of @ particles from a known weight of active 
material. A small quantity of the material under examina- 
tion was finely powdered in an agate mortar, and then mixed 
with alcohol or ether and deposited asa thin uniform film on 
a thin sheet of aluminium or glass. The method adopted 
was similar to that first used by McCoy. Care was taken 
that the powder suspended in the liquid was well stirred in 
order to avoid a separation of the lighter from the denser 
portions. The weight of the active film was determined by 
weighing the plate before and after the active material had 
been removed. It was desirable to use very thin films in 
order that all the « particles might emerge without much 
loss of their range. In the case of uranium, however, the 
number of « particles emitted was so small that they were 
difficult to count with accuracy. For this reason thicker 
films were in some cases purposely employed. The efficiency 
of the zinc sulphide screen was tested by counting the number 
of « particles emitted from a definite quantity of radium C. 
The number of scintillations observed was found to be 8 per 
cent. less than the actual number of « particles incident on 
the screen. The latter value was calculated from the known 
result that one gram of radium and each of its products 
emits 3°4 x 10'° « particles per second. In the initial experi- 
ments the number of scintillations was counted by placing 
the screen close to the active material. In this case, the 
number of « particles striking the screen is equal to one half 
the total number emitted from an area of the active film 
equal to the area of screen seen in the microscope. This 
method is open to some objections, for it requires that the 
film should be very uniformly spread and, in addition, very 
thin, for otherwise the particles emitted at an oblique angle 
suffer a considerable loss of range in the active material 
itself. The lack of uniformity of the film can be corrected 
for by counting at different points parts of the film, but this 
_ Involves much labour. 

In most of the experiments the active matter was spread 
in a circular area, and the small zine sulphide screen was 
placed parallel to the film and opposite to its centre. 


If a = radius of circular film, 
d = distance of screen from centre of film, 
A = area of screen observed in field of microscope, 
o = total number of particles emitted per second 
per square centimetre of surface of film, 


694 Dr. Geiger and Prof. Rutherford on the Number of 


then, by a simple integration, it can be shown that the 
number 2 of « particles incident per second on the area A is 
given by 


oA ad ) 
ee me 
Var+a@ 


A simple example will serve to illustrate the method of 
calculation. The uranium film No. 1 (see table later) con- 
tained 10°43 milligrams of uranium oxide (U3Qg) spread on 
an area of 5°9 square cms. 515 scintillations were counted, 
and the average number of scintillations observed corre- 
sponded to 5°16 per minute, and per second ‘086. Making 
the 8 per cent. correction for the imperfection of the screen, 
the corrected value becomes ‘093. This is the value of » to 
be substituted in the formula. 


A = 3°16 sq. mms. d = 2°06 ems. a'=' 1°37 cms. 
Substituting these values in the formula, 
a= 'a0'0. 


Now the weight of film per square centimetre was 
1°77 mg. U;O0,, or 1°50 mg. uranium. Consequently, from 
this experiment, the total number of « particles emitted per 
second per gram of uranium is 2°33 x 10°. 

The chief difficulty of the experiments lay in counting 
accurately a sufficiently large number of scintillations. The 
numbe~ 2f scintillations observed in the microscope varied 
from one to tive per minute in the case of uranium or 
thorium. While different observers agreed closely in 
counting scintillations due to radium or polonium when 
30 to 50 scintillations were seen per minute, the agreement 
was not so good for uranium films. This difference is in 
part due to the fact that the eye becomes quickly fatigued 
when only a few scintillations appear on the screen per 
minute. This was especially marked in counting the scintil- 
lations from uranium, which are relatively much fainter than 
those from radium ©. In the case of uranium and thorium 
minerals, where the scintillations are on the average much 
brighter than those from uranium, the counting was relatively 
easy. The brightness of scintillations of course depends on 
the range of the a particle striking the screen. We shall 
see later that the range of the a particle, and consequently 
the intensity of the scintillations from uranium, is less than 
from any other radioactive substance. 

The active materials used in these investigations were 


a Particles emitted by Uranium and Thorium. 695 


kindly presented to us by Professor Boltwood, and were 
fractions of larger quantities analysed by him. We desire 
to express our indebtedness to Professor Boltwood for the 
use of these materials. 

(1) Uranic-uranose oxide (U30,) prepared from uranium 
nitrate which had been crystallized fifteen times. The least 
soluble fraction was taken and ignited at a high heat ina 
current of oxygen. 

(2) Uraninite—a selected sample from Joachimsthal. This 
contained 61:7 per cent. of uranium. The mineral, when 
finely powdered, lost 6°2 per cent. of its emanation. The 
sample employed had been finely ground for several years, 
and during this time the emanation had steadily escaped. 
Under these conditions it can be simply deduced that the 
emission of « particles from the mineral is about three per 
cent. less than if the mineral had retained all its emanation. 
A correction of this amount has consequently been made to 
the counted number of « particles. 

(3) Thorium oxide prepared from thorite. This was tested 
five weeks after its chemical separation. Since, in the 
chemical process of purification, the mesothorium is removed 
from the thorium, the a-ray activity of the purified thorium 
decays with time due to the decay of its product radio- 
thorium. Since the half period of decay of the latter is 
about 737 days, a positive correction of about two per cent. 
is necessary to give the correct number of « particles emitted 
from thorium oxide in radioactive equilibrium. The activity 
of the thorium oxide in the form of a thin film was compared 
with that of a film of the mineral thorite of known composi- 
tion, and gave nearly the ratio to be expected from their 
relative content of thorium. 

The results of the observations are included in the following 
Table (p. 696). 

Since only about 900 scintillations were counted altogether, 
the agreement between the three uranium films is closer 
than could be expected, considering the possible errors in 
the experiment. In the case of the mineral films 2000 scin- 
tillations were counted in all, and about an equal number for 
the thorium films. Before and after each set of observations 
the screen was carefully tested to determine the number of 
scintillations observed when the active material was removed. 
The correction for the screen employed was small, and 
usually corresponded to one scintillation in three or four 
minutes. All the counting experiments were checked among 
themselves by measuring the activity of the films in an a-ray 
electroscope. The activity measured in this way was found 


696 Dr. Geiger and Prof. Rutherford on the Number of 


to be proportional to the weight of the film for thin films, 
but for the thicker films the activity was relatively smaller 
on account of absorption. 


| Number of & particles emitted 


Radioactive Substances. per second per gram of 
Uranium or Thorium. 


Uranium film No. 1. Fe ; 
10°43 mers. U0, on area 59 em.2 a 33x10 


Uranium film No. 2. : 4 Average 
2°85 mers. U,O, on area 12°8 em.” spe Ss 2°37 X 10# 
Uranium film No. 3. : 1 
3°04 mers. U,O, on area 14°9 cm.? 2:43 X10 
Mineral film No. 1. 
10°95 mgrs. Uraninite. 95 «10+ 
(Joachimstahl) on area 5°9 em.” 
pe ee Oe Tee : 1 Average 
} 9°6 A 104 
Mineral filin No. 2. 
12°73 mgrs. Uraninite. o7 x 10+ 
(Joachimstahl) on area 5°9 em.? 
Thorium film No. 1. EE 
4°43 mers. ThO, on 6:1 cm.,? aie Ls 
Thorium film No. 2 | A 
orium film No. 2. } verage 
1-21 mgrs. ThO, on 6°4 em.? | hats adh 277x108 
Thorium film No. 3. 2°65 x 10! 


3°58 mgrs. ThO, on 6°15 cm.? 


Tt will be seen that there is a good agreement between the 
experiments and the numbers calculated on the assumption 
considered in the beginning of this paper. This is brought 
out by the Table below. 


Number of a particles per gram 
of Uranium per second. 


Calculated. Observed. 
eae Oy Oe, 22 04 2°37 eee 
Uranium mineral......... 4°67 x 104 9-6 x104 


Thorium, number of particles per gram: 2°7 x10? 


No doubt the agreement is closer than would be expected 
under the conditions of the experiments. 


a Particles emitted by Uranium and Thorium. 697 


The agreement between theory and experiment confirms 
in another way the correctness of Boltwood’s conclusion that 
uranium emits two « particles for one from each of its later 
products. The experiments are not of sufficient accuracy to 
confirm the data on the relative activity of actinium and 
radium. There is no doubt, however, that the number of 
a particles to be ascribed to actinium is very small compared 
with that to be expected if actinium and its series of products 
emitted one « particle for one from radium. The connexion 
of actinium with the uranium-radium series is difficult to 
determine, and remains one of the chief outstanding problems 
in the analysis of radioactive changes. 


Production of Helium by Uranium, Uranium Minerals, 
and ‘thorium. 


Since the a particle is a charged atom of helium, it is a 
simple matter to deduce the rate of production of helium 
from the active materials considered. Calculation and 
experiment show that one gram of radium in equilibrium 
with its three «-ray products produces 158 cubic mm. of 
helium per year. Since radium and each of its products 
emits 3°4 x10" @ particles per gram per second, uranium, 
which emits 2°37x10* « particles per gram per second, 
produces 2°75x 107° cubic mm. per year. The rate of 
production of helium for the different materials is given 
below. 


Production of Helium 
per gram per year. 


Deere... ws 'cvrtcaie eed 2x Om cubic mine 
JUNG aera eee ee Bie aa ae LOr? 3) 
Uranium mineral in equi- ; me 

et, EO Sel s 

MO CLIN, cia gcte asd Hee 
Radiam in equilibrium ... 158 - 


A simple calculation allows us to estimate the production 
of helium for a mineral like thorianite containing both 
uranium and thorium. 


Range of the « particles from Uranium. 


The range of the « particles from uranium has been 
difficult to determine directly on account of the smallness of 
the activity of the thin films of the substance. By observa- 
tions of the decrease of the ionization due to a layer of 
uranium when sheets of thin aluminium were placed over it, 
Bragg * deduced that uhe range in air of the e particle from 
uranium was about 3°5 cms. In the course of counting the 
scintillations from a thin film of ionium, it was observed that 

* Brage, Phil. Mag. 1906, xi. p. 754. 


toler) 


698 Prof. E. Rutherford and Dr. H. Geiger on the 


the scintillations were as bright if not brighter than those 
from a thin film of uranium. Boltwood has found that the 
range of the « particle from ionium is 2°8 cms., so that it 
appeared probable that the range of the « particles from 
uranium had been overestimated. This conclusion was 
confirmed by finding that the « rays from a thin film of 
uranium were more readily absorbed by aluminium than 
those from ionium. By a special method, the range of the 
a particle from uranium has been measured and found to be 
about 2°7 ems., while the range of the « particle from ionium 
is a millimetre or two longer. Further experiments are in 
progress to determine the range of the e particle from 
uranium accurately, and to examine carefully whether two 
sets of « particles of different range can be detected. 


University of Manchester, 
July 1910. 


LXXVI. The Probability Variations in the Distribution of 
a Particles. By Professor EH. RuruErrorp, /.2.S., and 
H. Geicer, PhD. With a Note by H. Bateman *. 

ie counting the « particles emitted from radioactive 

substances either by the scintillation or electric method, 

it is observed that, while the average number of particles 
from a steady source is nearly constant, when a large number 
is counted, the number appearing in a given short interval 
is subject to wide fluctuations. These variations are especially 
noticeable when only a few scintillations appear per minute. 
For example, during a considerable interval it may happen 
that no « particle appears ; then follows a group of «& par- 
ticles in rapid succession ; then an occasional « particle, and 
soon. It is of importance to settle whether these variations 
in distribution are in agreement with the laws of probability, 
2. e. whether the distribution of « particles on an average is 
that to be anticipated if the & particles are expelled at random 
both in regard to space and time. It might be conceived, 
for example, that the emission of an e particle might pre- 
cipitate the disintegration of neighbouring atoms, and so 
lead to a distribution of « particles et variance with the 
simple probability law. 

The magnitude of the probability variations in the number 
of « particles was first drawn attention to by H. v. Schweidler fT. 
He showed that the average error from the mean number of 
a particles was /N .t, where N was the number of particles 
emitted per second and ¢ the interval under consideration. 
This conclusion has been experimentally verified by several 


* Communicated by the Authors. We es 
+ v. Schweidler, Congrés Internationale de Radiologie, Liéve, 1905. 


ee 


Probability Variations in Distribution of « Particles. 699 


observers, including Kohlrausch*, Meyer and Regener f, 
and H. Geiger ft, by noticing the fluctuations when the 
ionization currents due to two sources of « rays were balanced 
against each other. The results obtained have been shown 
to be in good agreement with the theoretical predictions of 
von Schweidler. 

The development of the scintillation method of counting 
a particles by Regener, and of the electric method by 
Rutherford and Geiger, has afforded a more direct method 
of testing the probability variations. Examples of the dis- 
tribution of « particles in time have been given by Regener $ 
and also by Rutherford and Geiger ||. 1t was the intention 
of the authors initially to determine the distribution of 
a particles in time by the electric method, using a string 
electrometer of quick period as the detecting instrument. 
Experiments were made in this direction, and photographs 
of the throws of the instrument were readily obtained on a 
revolving film; but it was found to be a long and tedious 
matter to obtain records of the large number of @ particles 
required. It was considered simpler, if not quite so accurate, 
to count the « particles by the scintillation method. 


Experimental Arrangement. 


The source of radiation was a small disk coated with 
polonium, which was placed inside an exhausted tube, closed 
at one end by a zine sulphide screen. The scintillations 
were counted in the usual way by means of a microscope on 
an area of about one sq. mm. of screen. During the time 
of counting (5 days), in order to correct for the decay, the 
polonium was moved daily closer to the screen in order that 
the average number of « particles impinging on the screen 
should be nearly constant. The scintillations were recorded 
on a chronograph tape by closing an electric circuit by hand 
at the instant of each scintillation. Time-marks at intervals 
of one half-minute were also automatically recorded on the 
same tape. 

After the eye was rested, scintillations were counted from 
3 to 5 minutes. The motor running the tape was then 
stopped and the eye rested for several minutes ; then another 
interval of counting, and so on. It was found possible to 
count 2000 scintillations a day, and in all 10,000 were 
recorded. The records on the tape were then systematically 

* Kohlrausch, Wiener Akad. exv. p. 673 (1906). 

+ Meyer and Regener, Ann. d. Phys. xxv. p. 757 (1907). 

t Geiger, Phil. Mag. xv. p. 539 (1908). 

§ Regener, Verh. d. D. Phys. Ges. x. p. 78 (1908); Sitz. Ber. d. K. 


Preuss. Akad. Wiss. xxxviii. p. 948 (1909). 
|| Rutherford and Geiger, Proc. Roy. Soc. A. Ixxxi. p. 141 (1908). 


700 Prof. E. Rutherford and Dr. H. Geiger on the 


examined. The length of tape corresponding to half-minute 
marks was subdivided into four equal parts by means of a 
celluloid film marked with five parallel lines at equal distances. 
By slanting the film at different angles, the outside lines 
were made to pass through the time-marks, and the number 
of scintillations between the lines corresponding to 1/8 minute 
intervals were counted through the film. By this method 
correction was made for slow variations in the speed of the 
motor during the long interval required by the observations. 

In an experiment of this kind the probability variations 
are independent of the imperfections of the zinc sulphide 
screen. The main source of error is the possibility of missing 
some of the scintillations. The following example is an illus- 
tration of the result obtained. The numbers, given in the 
horizontal lines, correspond to the number of scintillations 


for successive intervals of 7°5 seconds. 
Total per minute. 


determi: Sco ad 6h | AO 200) Kier 25 
ana. %, Bigne et Gy Al ta ee 30 
Mt er Bh As theca pile dds days neha 24 
AEs "4-95 Bi Bsieay ed eed ae OL 
alma! 555 G A216 (Eye ho® ya ee 42 
Average for 5 minutes... 30°4 
Erie average ivi Aoi 31°0 


The length of tape was about 14 cms. for one minute 
interval. The average number of particles deduced from 
counting 10,000 scintillations was 31:0 per minute. It will 
be seen that for the 1/8 minute intervals the number of 
scintillations varied between 0 and 10; for one minute 
intervals between 25 and 42. 

The distribution of « particles according to the law of 
probability was kindly worked out for us by Mr. Bateman. 
The mathematical theory is appended as a note to this paper. 
Mr. Bateman has shown that if w be the true average number 
of particles for any given interval falling on the screen from 
a constant source, the probability that n a particles are 


. . . . ic . 
observed in the same interval is given by —,e-*. 1m is here 
n . 


a whole number, which may have all positive values from 
0 toc. The value of w is determined by counting a large 
number of scintillations and dividing by the number of 
intervals involved. The probability for n a@ particles in the 
given interval can then at once be calculated from the theory. 
The following table contains the results of an examination of 
the groups of @ particles occurring in 1/8 minute interval. 


: 
| 
: 
: 


f « Particles. TOL 


zon O 


Distribut 


lity Variations in 


Probab 


para | 0 15 2 824 8 6 fe 


& particles. 


I. cae, .| 15" SOS106 3152 170-102 238 50] Wie ae 3179 
Tl; veo. 172 89 $8 4116 120 208 0a. of) eee! 2334 
Tih. & | 15. 56= 07 189 21s 206 G02 2G= ee se eg eee 2373 
LV ..i.390| 102 52-02 116 124 292 G2" 262 *Ge es 0 eee 2211 
Sum ....| 57 208 883 525 582 408 978 139 45-27 10 4° 0 i e007 


ie 54 210 407 525 508 394 254 140 68 29 11 41 4 1 


values. 


intervals. 


792 
596 
632 
588 


2608 


Number of | Number of | Average 
a particles. 


number. 


4°01 
3°92 
3°75 
3°76 


3°87 


702 Prof. E. Rutherford and Dr. H. Geiger on the 


For convenience the tape was measured up in four parts, 
the results of which are given separately in horizontal columns 
I. to IV. 

For example (see column I.), out of 792 intervals of 
1/8 minute, in which 3179 @ particles were counted, the 
number of intervals giving 3 a particles was 152. Combining 
the four columns, it is seen that out of 2608 intervals containing 
10,097 particles, the number of times that 3 a particles were 
observed was 525. The number calculated from the equation 
was the same, viz. 525. It will be seen that, on the whole, 
theory and experiment are in excellent accord. The difference 
is most marked for four « particles, where the observed number 
is nearly 5 per cent. larger than the theoretical. The number 
of « particles counted was far too small to fix with certainty 
the number of groups to be expected for a large value of n, 
where the probability of the occurrence is very small. It 
will be observed that the agreement between theory and 
experiment is good even for 10 and 11 particles, where the 
probability of the occurrence of the latter number in an 
interval is less than 1 part in 600. The closeness of the 
agreement is no doubt accidental. The relation between 
theory and experiment is shown in fig. 1 for the results given 
in Table I., where the o represent observed points and the 
broken line the theoretical curve. 


r Fig. 1. 
CG 
500 ma 
(eo) 

400 
% 
S 
3300 
SS [o) 
S 
S200 ° 

0. 
100 


° 


ne 
\ “oo 


e 4 6 8 10 l2 


NUMBER OF & PARTICLES IN INTERVAL 


—_ 


The results have also been analysed for 1/4 minute intervals. 
This has been done in two ways, which give two different 
sets of numbers. For example, let A, B, C, D, E represent 
the number of « particles observed in successive 1/8 minute 
intervals. One set of results, given in Table A, is obtained by 
adding A+B, C+D, &c. ; the other set, given in Table B, 


TABLE A, 


ee 


| Whole Whole Average 


703 


x aaa | 0-12-38 4 5.6 272.8 99-10 de 12 13 db aGaAly see number cf | number of | number in 
es elt a scintillations.| intervals. | one interval. 
© Leven) 0°58 4.721% 85-42 60271 29546 22-10 lie eee Oe 3182 396 8:04 
= 1T.2.....| 0 2 9 %6-19 88594 56-88 BioaT Baal 7 ap = lt ee 2330 298 7°82 
| TID ..scss| 0 0G B26 BO 89 $1442) 08 SoG ay Sit ties ee EU ee) 2373 316 751 
S TV .ccccx..| 0-4 7 VLA: 80240 47248 OSE TUB ee et ee 2214 294 7°53 
3 Sum ......| 0 6 20 32 75 187 155 214 198 157 126 81 49281810 4 4001 0 10099 1304 774 
RS 

a TABLE B. 

2 Lcc| 0-2 4 929k 35346 BOces deseo G0 lb 1? 6 eed 0 0 0, 0 3180 396 8:03 
2 il... lO 1 3 OSU Oy S08 ae S408 Se ee OL og 0 oe 2333 298 7°83 
S| TIL..c.| 0.0 12 88 88932 Piles f0eS2 Wt lt Gopal 3020 0 0 01 2371 316 7°50 
S TY ....5. ;|.0 0 8 18-08 26295 44-56 w7ese tb 6 6a 0 120. 20 bo 2210 294 752 
iS 1 gum ....10 9 17 46 90°126.151 187180 179 181 75 44 86 18 14-1 1 2 4 1 7D 10094 1304 774 
= 

< Tee ceRt| O 9 87 78 174 263 306 401 373 330 257 156 93 63 2924 5 6 21 2 I 
er 20193 2608 774 
a oe Li1-1 9 84 88 170 263 339 372 863 312 242 170 110 65 36.19 9 4 18-72-28 10 


| | 


fn ———— 


704 Mr. H. Bateman on the 


by starting 1/3 miaute later and adding B+C, D+H, Ke. 
The results are given in the appended Tables. In the final 
horizontal columns are given the sum of the occurrences in 
Tables A and B and the corresponding theoretical values. 

In the cases for 1/4 minute intervals, the agreement between 
theory and experiment is not so good as in the first experi- 
ment with 1/8 minute interval. It is clear that the number 
of intervals during which particles were counted was not 
nearly large enough to give the correct average even for 
the maximum parts of the probability curve, and much less 
for the initial and final parts of the curve, where the pro- 
bability of an occurrence is small. However, taking the 
results as a whole for the 1/8 minute and the 1/4 minute 
intervals, there is a substantial agreement between theory 
and experiment, and the errors are not greater than would 
be anticipated, considering the comparatively small number 
of intervals over which the «& particles were counted. 
We may consequently conclude that the distribution of 
« particles in time is in agreement with the laws of pro- 
bability and that the « particles are emitted at random. As 
far as the experiments have gone, there is no evidence that 
the variation in number of « particles from interval to 
interval is greater than would be expected in a random 
distribution. 

Apart from their bearing on radioactive problems, these 
results are of interest as an example of a method of testing 
the laws of probability by observing the variations in 
quantities involved in a spontaneous material process. 


University of Manchester, 
July 22nd, 1910. 


Note. 


On the Probability Distribution of « Particles. 
By H. Bateman. 


Let dt be the chance that an a particle hits the screen in a 
small interval of time dt. If the intervals of time under 
consideration are small compared with the time period of the 
radioactive substance, we may assume that is independent 
of t. Now let W,(¢) denote the chance that n @ particles hit 
the screen in an interval of time ¢, then the chance that 
(n+1) particles strike the screen in an interval ¢+dt is the 
sum of two chances. In the first place,n+1 « particles may 
strike the screen in the interval ¢ and none in the interval dé. 
The chance that this may occur is (1—Adt)Wn4+1(é). 
Secondly, n « particles may strike the screen in the interval ¢ 


Probability Distribution of « Particles... 7095 
and one in the interval dt ; the chance tha this may occur is 
AdtW,,(t). Hence 

Wrsi(t+dt) = (1—drdt) Wrarlt) + ndtW(2). 


Proceeding to the limit, we have 


dW, | Rea e 
eels ce = A(W,,— ‘ar ae 


Putting n=0, 1, 2... in succession we have the system 
of equations : . 


dW 
rc —rW,; 
dW 
dt = ant ACWo— Aids 
dW AXT 
de a ACW, —W,), 


which are of exactly iba same ‘ont as those occurring in thé 
theory of radioactive transformations *, except that the time- 
periods of the transformations would have to be assumed to 
be all equal. 

The equations may be solved by ae each of them 
by @* and integrating. Since WO) so eth ) =0, we 


have in succession : 
q Wo — e—*t ; ‘ : 
dt (We) = 7 ur W,= Kier, : | m 


d MOP 
di (W,¢**) = A, he Wea a : Ans 


and soon. Finally, we get 


W,= 7h et 


The average number of @ particles which strike the screen 
in the interval ¢ is A¢. Putting this equal to 2, we see that 
the chance that n « particles strike a screen in this * 
interval is \ 


W,. — OT. 
nv 


* Rutherford, ‘ Radioactivity,’ 2nd edition, p. 330. The chance that, 
an atom suffers m disintegrations in an interval of time ¢ is equal to the 
ratio of the amount of the nth product present at the end of the interval 
to the amount of the primary substance present at the commencement, 


Phil. Mag. 5: 6, Volu20; No. 18. Océ: 1910. 3 A 


706 = Onthe Probability Distribution of « Particles. 


The particular case in which n=O has been known for 
some, time (Whitworth’s ‘Choice and Chance,’ 4th ed. 
Prop. 01). 

If we use the above analogy with radioactive trans- 
formation, the theorem simply tells us that the amount of 
eur substance remaining after an interval of time ¢ is 

# if a unit quantity was present at the commencement. 

The probable number of « particles striking the screen in 

the given interval is 


m—1 
pS Ween = we 


ni if ml (21 ip. 


The most probable number is obtained by finding the 
maximum value of Wn. 


nr 


Since ; = ~ this ratio will be greater than 1 so long 


n—-l 
asn<wx. Hence if nS 2, 


W, = Wiest 3 


if n=27, Wiz=W,-1. The most probable value of n is 
therefore the integer next greater than « ; if, however, « is 
an integer, the numbers e—1 and wx are equally probable, 
and more probable than all the others. 

The value of X% which is calculated by counting the total 
number of a particles which strike the screen in a large 
interval of time T, will not generally be the true value of 2X. 
The mean deviation from the true value of 2 is calculated 
by finding the mean deviation of the total number N of 
a particles observed in time T from the true average number 
AT. This mean deviation D (mittlerer Fehler) is, according 
to the definition of Bessel and Gauss, the square root of the 
probable value of the square of the difference N—AT, and so 
is given by the series 


D = > (N-ATy ae goat 
oe [OD | ODF Ont | oDiare 
» Slat? (N—D!-W_1 +, aoe Jaa 


Hence D=,/nXT, and the mean deviation from the value 


A New Radiant Emission from the Spark. T07 


of X is accordingly Ay 

D r 

a ee 
it thus varies inversely as the square root of the length of 
the interval of time. ‘This result is of the same form as the 
classical one used by E. v. Schweidler in the paper referred 
to earlier. 

The probable value of | N—AT | (der durchschnittlicher 

Fehler) is much more difficult to calculate. 


LXXVII. A New Radiant Emission from the Spark. By 
R. W. Woop, Professor of Hxperimental Physics in the 
Johns Hopkins University *. 


[Plate XIV. ] 


SCARCELY know how to designate the peculiar type 
of radiation referred to in the present paper, which 
was first discovered over two years ago in the course of some 
experiments made with a view of ascertaining whether the 
Schumann waves from the spark gave rise to any fluores- 
cence of the air by which they were absorbed. It is now 
known that there is a feeble ultra-violet luminosity of air or 
nitrogen gas surrounding a small mass of radium, in other 
words the radium renders the gas luminescent. To test fer 
a fluorescence due to the absorption of very short light- 
waves, the condenser spark between aluminium electrodes 
was passed behind and very close to a vertical strip of metal 
which completely concealed the spark, but which enabled 
observation, either visual or photographic, of the air in its 
immediate vicinity. If the air in the room was free from 
dust and smoke absolutely nothing could be seen with the 
eye, even after prolonged resting in the dark. A photo- 
graph, however, made with a smail camera provided with a 
quartz lens, showed that the air around the spark was a 
source of a powerful actinic radiation, which was completely 
stopped by the intervention of a glass plate between the 
camera and the spark. The first photograph of the pheno- 
menon which was obtained is reproduced on Pl. XIV. fig. 6. 
The narrow strip of metal between the two wider strips was 
about 1 cm. in width; the spark discharge was concealed 
behind this. 
T'wo hypotheses immediately presented themselves : (a) we 
are dealing with a scattering of the shortest waves by the 


* Communicated by the Author. 
a A 2 


708 Prof. R. W. Wood on a New 


air molecules or microscopical dust particles, as in case of, 
blue-sky ; (6) ultra-violet fluorescence of the air caused by 
the absorption of the Schumann waves. As a matter of fact 
neither hypothesis turned out to be tenable, but I mention 
them to show that they have been carefully considered, 

If the emission of ultra-violet light by the air was merely 
a scattering, its spectrum should be identical with that of the: 
spark : if, on the contrary, it is a fluorescence phenomenon,, 
its spectrum would be totally different. In the spectro- 
scopic work it was necessary to get as close as possible to the 
spark, and yet run no risk of having its direct light enter 
the slit of ‘the instrument. To meet this requirement tlie 
apparatus shown in fig. 1 was constructed. <A disk of 


Fig. 1. 


aluminium, 3 mm. thick, was perforated with a hole 1°5 mm.,. 
in diameter and fastened to a short brass cylinder B. The. 
aluminium electrode C was carried by a screw D, which. 
passed through the ebonite cap E. The spark « discharge: 
passed between C and the inner rim of the hole in the. 
aluminium disk. If the spark chamber is hermetically. 
sealed the explosive expansions of the air are apt to force 
the spark aureole, which is pale green in the case of the 
aluminium spark, out through the hole. The small lateral: 
tube J prevents this, and serves as well for the introduction 
of various gases. The length of the spark must be so. 
adjusted that no visible portion is forced out through the 
hole, when viewed from the position K. } 

A second brass cylinder F, closed at the top and fitted 


Radiant Emission from the Spark. 709 


with two lateral tubes H and G, can be screwed to the spark 
chamber when it is desired to study the emission in dry 
filtered air or some other gas. The tube G is closed with a 
quartz window, while H terminates ina smaller tube J for the 
introduction of gas. The emission is quite invisible in dust- 
free air, yet it can be photographed with an exposure of one 
or two minutes with a quartz lens of 2 cm. aperture and 
15 cm. focus. In arranging the position of the quartz 
camera the focal plane should be examined with an eyepiece 
in a dark room to make sure that no part of the lens receives 
light from the edge of the hole ; in other words, the top of the 
lens must be just below the plane of the aluminium disk A. 
In order to get an idea of the appearance of scattered 
light, the air around the apparatus was filled with smoke and 
the spark discharge started. The photograph obtained in 
this way is reproduced on Pl. XIV. fig. 1. Fig. 2 was 
obtained when the air was free from smoke or dust, and 
shows the appearance of the emission with which we are 
concerned. A comparison of these two photographs shows 
us at once that the emission does not extend nearly so far 
out from the aperture as does the luminous region of light- 
scattering smoke. It appears as if it were rapidly absorbed 
by the air. That this is not due to differences in the time of 
exposure is shown by the original negatives, for fig. 2 has a 
density nearly double that of fig. 1 in the immediate proximity 
of the aperture in the disk. An experiment was next made 
to ascertain the nature of the light given out by the emission. 
A fine thread of fused quartz, about 2 mm. in length, was 
mounted at the edge of the aperture by means of a small 
arop of soluble glass. This scattered the light of the spark, 
forming a narrow linear source of spark light located at the 
centre of the base of the emission. The slit tube of a small 
quartz spectrograph was removed, and the luminous quartz 
thread brought into its place. The resulting photograph is 
shown in fig. 3, a continuous band of light, the spectrum of 
the quartz fibre, with the emission above it and about at its 
centre. ‘his picture proved that the light given out by the 
emission embraced a limited range of wave-lengths in the 
region 300-310. This picture was secured with an exposure 
of only fifteen minutes, which made it seem probable that the 
spectrum of the emission could be obtained with a fairly 
narrow slit. Fig. 5 shows a spectrogram obtained with a 
wide slit, the aluminium lines showing faintly as a result of 
ditfused light: the lower spectrum is that of the spark for 


‘comparison. The spectrum of the emission consists of two 
‘broad bands, one very strong, the other (to the right) much 


710 Prof. R. W. Wood on a New 


weaker. These were found to be identical with the so-called 
“water bands” of the oxy-hydrogen flame, as is clearly 
brought out by fig. 9, in which the upper spectrum is that 
of the oxy-hydrogen flame, the lower that of the emission. 
In addition to these bands I obtained on one plate lines at 
wave-lengths 3576, 3537, and 3369, which are identical with 
lines attributed to nitrogen in the spectrograms published 
by Eder and Valenta of the spark between wet carbon 
electrodes. There is, in addition, a line which is imbedded 
in the water-band, as shown in figs. 7 and 8. In fig. 7 the 
upper spectrum is that of the oxy-hydrogen flame (over- 
exposed), below it the aluminium spark, and at the bottom 
the spark emission. The nitrogen lines come out very clearly 
in this case. The spectrum by Eder and Valenta, which is 
practically identical with that of the emission, was obtained 
by passing the discharge of an induction-coil between wet 
carbon electrodes, and differs from that of the oxy-hydrogen 
flame in that it shows the nitrogen lines above referred to. 

It looked very much as if the emission might be due to 
the fluorescence of nitrogen and water vapour, resulting 
from the absorption of the Schumann waves ; this would 
explain its failure to penetrate the air to any considerable 
distance. ‘To test this point the auxiliary tube was attached 
to the spark chamber, the emission being studied through 
the quartz window attached to the tube G. The apparatus 
was first filled with air carefully dried by passage through a 
tube filled with phosphorus pentoxide, and then with air 
passed through a plug of wet cotton. The emission was 
photographed in each case, but no difference in the intensity 
of the images could be detected. Oxygen and nitrogen were 
then tried in succession. In the former there was almost no 
trace of the emission, while in the latter it was much brighter 
and extended toa greater distance trom the aperture than 
in air. Photographs of the phenomenon in these two gases 
are reproduced on Pl. XIV. fig. 4. The emission is photo- 
graphed against the very black background furnished by the 
long tube H, in fig. 1. The time of exposure was the same 
in each case, and the two plates were developed together. 
The aperture is to the right in each picture, the emission 
shooting out towards the left. The crescent of light is the 
inner edge of the tube H illuminated by diffused light. 

The next question was to determine whether the presence 
of oxygen prevented the formation of the emission, or - 
whether the gas exerted an absorbing action. This was a 
difficult matter to determine, since numerous experiments 
showed that no substance was transparent to the emission. 


Radiant Emission from the Spark. 711 


A plate of white fluorite, 0°5 mm. in thickness, which had 
been found very transparent for the Schumann waves by 
Dr. Lyman, who very kindly placed it at my disposal, 
together with an end-on hydrogen tube for the production of 
Schumann waves, when placed over the aperture was found 
to destroy all trace of the emission. This disposed of the 
theory that we were dealing with a fluorescence produced by 
the short waves. Thin aluminium foil, such as is used with 
the Lenard tubes, was found to be equally opaque. It is 
therefore a difficult matter to start the emission in a given 
gas and pass it into a different one. The problem was finally 
solved by an experiment designed to test one of the theories 
that I had evolved to explain the phenomenon. It occurred 
to me that we might be dealing with hydrogen ions, shot off 
from the electrodes, which, by combination with the oxygen 
of the air, gave rise to a spectrum similar to that of the oxy- 
hydrogen flame. We might in this way explain the lessened 
effect in oxygen as a result of the circumstance that the 
‘combustion ” of the ions took place almost entirely within 
the small tube with which the disk of aluminium was per- 
forated. If this were the case, it seemed probable that if 
the emission were formed in air, and a small jet of oxygen 
were directed across it transversely, we should observe a 
more intense action at the point where the emission met the 
oxygen jet. The experiment was tried, and it was found 
that the gas jet merely interrupted the emission, killed it in 
other words, precisely as if it absorbed it. If the emission 
was started in air and a jet of nitrogen blown gently against 
the aperture, the emission was found to shoot out much 
farther and to be of greater intensity. The magnetic field 
appeared to be without action on it, though the experiment 
was found to be attended with difficulties on account of the 
action of the magnet on the spark. 

It is still more difficult to study the action of an electro- 
static field. The material constituting the emission is 
evidently shot from the aperture at a very high velocity, for 
it is impossible to blow it aside with a strong jet of air ; 
moreover, if air is forced continuously into the auxiliary 
chamber, passing through the aperture in the aluminium 
disk at a high velocity, the emission does not appear to be 
held back in the slightest degree. 

I am unable to explain its reactions with oxygen and 
nitrogen, and the apparent failure of the presence or absence 
of water vapour to modify the intensity of the spectrum, 
which is made up chiefly of the so-called water-bands. 
These bands appear when hydrogen burns in oxygen, and yet 


712) Prof. R. W. Wood : Some Experiments on 


oxygen destroys the luminosity of the emission. This fact 
appears to be of the greatest importance in connexion with 
the origin of these bands. If I remember rightly, the intro- 
duction of chlorine gas into a sodium flame destroys its 
emission of the D lines, and there may be some analogy 
between the two phenomena. I intend sometime to photo- 
graph the spark directly utilizing the principle of the spectro- 
heliograph. An image of the ‘spark obtained with mono- 
ehromatie light of the wave-length of the water-band may 
tell us something about the origin of the emission. In the 
meantime I hope that some study of the phenomenon will 
be made by others, as it appears to be of considerable im- 
portance in connexion with the origin of radiation. 

It seems quite likely that the ‘‘ Entladungsstrahlen ” 

be identical with the emission, for they are absorbed by 
oxygen. One great difficulty in the investigation is the 
apparent impossibility of separating the emission from the 
ultra-violet and visible light which goes out with it. 


‘ LXXVIII. Some Experiments on Refraction by 
non-homogeneous Media. By R. W. Woop* 


[Plate XIV. fig. 10. ] 
mR the apparent diameter of a body surrounded by a 


refracting atmosphere is slightly larger than its true 
diameter is well known. An extreme case is the mercury 
thread of a thermometer. At the other extreme we have the 
earth as seen from the moon. 

This magnification by a non-homogeneous atmosphere, in 
which there is no sharply defined refracting surface (as in 
the case of the earth’s atmosphere) can be very nicely shown 
in the following way :— 

Make a small rectangular glass tank by cementing five 
squares of glass together with sealing-wax. [ill it with 
melted gelatine and support an empty test-tube in the fluid 
with a clamp stand. ‘The bottom of the test-tube should be 
within half a centimetre of the bottom, After the jelly has 
solidified, pour hot water into the test-tube, and immediately 
withdraw it. It will leave a cylindrical hole in the jelly, 
with a hemispherical bottom. Now pour a mixture of 
glycerine and powdered chalk into the cavity until it is half 
full. Fill the remainder with water to which a few drops 
of milk have been added. The glycerine will gradually 


* Communicated by the Author. 


Refraction by non-homogeneous Media. 713 


diffuse into the gelatine, increasing its refractive index. 
The condition at the end of a few minutes will be not unlike 
that of a white body surrounded by a dense atmosphere, for 
the refractive index will be high at the boundary between 
the jelly and glycerine, gradually decreasing as we pass out 
into the jelly. The magnification resulting can be seen by 
looking through the side of the trough, the lower portion of 
the cavity appearing swollen out like a mushroom. If we 
perform the experiment with pure glycerine and clean water 
the same thing happens. By placing an are light behind 
the tank and throwing an image of the cavity upon a piece 
of ground glass with a camera objective, placed at the centre 
of the shadow of the tank, we can see the bright ring of 
light which appears to surround the bottom of the cavity. 
This is analogous to the ring of light which would be seen 
surrounding the earth by an observer on the moon during a 
lunar eclipse, or rather a solar eclipse. As the glycerine 
penetrates into the jelly this ring of light eventually separates 
from the line of the cavity. Photographs of this experiment 
are reproduced in fig. 10, Plate XIV. 

Hxner has described experiments with pseudo-lenses made 
by immersing gelatine cylinders in water, and drying 
sections of gelatinous cylinders. These I described in 
‘Physical Optics,’ but have since improved the method by 
the use of glycerine. The whole experiment can now be 
performed within the limits of the lecture hour. 

A handful of photographic gelatine is soaked in clean 
water until thoroughly softened. The excess of water is 
poured off and the mass is then heated until quite fluid, and 
tiltered through a funnel with a small piece of absorbent 
cotton placed at the bottom of the cone. If the gelatine 
refuses to run through, add a little more boiling water. 
Pour a small quantity into a test-tube, and let it stand until 
solid. Hvaporate the remainder over a small flame, stirring 
constantly until it is of the consistency of syrup. This 
means boiling it down to one-third or less of its original 
volume. Now add an equal volume of glycerine, and pour 
the mixture into a second test-tube. After the jellies have 
set, crack the bottom of the tubes by a sharp blow, warm 
them by the momentary application of a Bunsen flame and 
push out the cylinders. 

Cut the cylinders into disks of different thicknesses, with 
a warm pen-knife. The best thickness is about two-thirds 
of the diameter. Mount the disks between small squares of 
thin plate-glass (window glass will do), warming the plates 
slightly, to insure getting the jelly into optical contact. 


714 Dr. G. W. C. Kaye on a Method of 


It may be found necessary to prop the upper plate in position 
until the surface in contact with the glass has “set.” The 
cylinders which are made of gelatine and water are now to 
be immersed in glycerine, the glycerine jelly cylinders in 
cold water. The glycerine should be stirred occasionally, 
as the layers in contact with the jelly take up the displaced 
water. ‘The action will be found to be well under way in a 
quarter of an hour, the glycerine gradually diffusing into 
the jelly, driving out the water, and the water gradually 
replacing the glycerine. A jelly containing glycerine has a 
higher refractive index than one containing water, con- 
sequently the cylinders soaked in glycerine act as concave, 
while those soaked in water act as convex lenses. 

The focal length wili be found to be only 8 or 10 ems., 
and very sharp images of the filament of an incandescent 
lamp or a gas-flame can be obtained with them. 

Interesting refraction effects can be observed by nearly 
closing the ends of a tin pipe 3 or 4 metres long and 10 or 
15 cms. in diameter with plate glass, inclining the tube and 
pouring in sufficient gasolene (petrol) to wet the entire 
bottom of the tube. On tilting the tube back into the 
horizontal position, the cross section of the circular end 
appears deformed into an ellipse when viewed through the 
opposite end with the eye near the bottom, and external 
objects are seen much distorted. Proximity of a flame is to 
be avoided. 


LXXIX. On a Method of Counting the Rulings of a Dyf- 
fraction Grating. By G.W. C. Kayes, B.A., D.Sc. The 
National Physical Laboratory *. 


[Plate XV. |] 


T would appear from a review of the earlier determinations 
of wave-lengths by the use of diffraction gratings, that 
most of the results were vitiated by an imperfect knowledge 
of the value of the grating-space rather than by inaccurate 
measurement of angular deviation. 

This was the case with the pioneer work of Fraunhofer 
(1814-1823) with wire and later with glass gratings; of 
Ditscheiner (1864, 1866) with one of Fraunhofer’s gratings ; 
and of Mascart (1864), Angstrém (1864), and van der Wil- 
ligen (1868), each of whom worked with Nobert’s gratings fT. 


* Communicated by Dr. R. T. Glazebrook, F. B.S. 
+ For a bibliography see Bell, Phil, Mag. xxv. p. 250 (1888). 


Counting the Rulings of a Diffraction Grating. 715 


Most of Nobert’s gratings, however, were small and inaccu- 
rately ruled, they all gave very imperfect definition and 
showed numerous “ ghosts.” There does not seem to have 
been any special trouble in ruling lines as close together as 
need be; for example Nobert, who jealously guarded his 
machine and methods as a trade secret, succeeded in ruling 
as many as 100,000 lines to the inch. The real difficulty 
was to secure uniformity of spacing. 

It was about forty years ago that Rutherfurd, a New York 
lawyer, by attention to the accuracy of the feeding-screw of 
his ruling machine, was able to make a great advance in the 
art of ruling gratings. The best of Rutherfurd’s gratings, 
however, were still faulty in respect of uniformity of spacing, 
and his larger gratings are not satisfactory. 

As is well known, Rowland’s success at Baltimore in 
ruling gratings was largely attendant on the success of his 
method to secure perfection in his feeding-screw. As at 
present made, the Rowland gratings are usually ruled with 
10,000, 14,438, or 20,000 lines to the inch. Practically all 
are on speculum metal (¢. 7 Cu, 3 Sn) which, permitting a 
high polish, yields gratings of great brilliance and definition, 
and being soft is not severe on the ruling diamond. On the 
other hand, speculum metal is not a simple alloy—a state of 
things which tends to local heterogeneity—itis heavy (which 
necessitates thick and rather massive gratings to prevent 
distortion), and further it has a considerable thermal coeffi- 
cient of expansion (19°3 x 10-°)—a fact which introduces 
some uncertainty into the certified grating-space for those 
gratings whose temperature of ruling is unknown. 


A determination of the grating-space involves (1) the 
measurement at a known temperature of the overall length 
of a selected number of rulings, (2) the counting of those 
rulings. The first part presents no difficulty for the modern 
comparator fitted with suitable high-power micrometer 
microscopes. It was to carry out what threatened to be a 
hepelessly tedious and fatiguing task in the counting of the 
rulings that the following method was employed in the case 
of a plane Rowland grating about 8 cms. long on speculum 
metal, belonging to Mr. J. W. Gifford, and for which 
ignorance of the ruling temperature made the certified value 
(14,438 lines to the inch) not so certain as was required. 

With some of Rowland’s gratings every fiftieth and 
hundredth rulings are differentiated by being shorter and 
longer than the rest. Others have all the rulings the same 
length ; this was the case with Mr. Gifford’s grating, 


716 ~ Dr. G. W. C. Kaye on a Method of 


Accordingly, quite close up to one edge of the grating 
were ruled by the Laboratory dividing-engine short fine 
equidistant reference-lines each about 4 mm. long and 
at such a distance apart as to include abou 100 crating 
lines. 

Hach fifth reference-line was a trifle longer than its 
neighbours, and each tenth line longer still. Byery fiftieth 
line was distinguished by the addition of an appropriate 
number of fine dots, so that afterwards there was very little 
trouble in picking up under the microscope any particular 
reference-line required. 

The counting of the grating rulings was carried out with 
the aid of the projection microscope. The grating was 
mounted on the stage of the microscope and an image of the 
graduated edge of ‘the erating was thrown upon a screen : 
with the magnific ation of 1200 employed the lines stood out 
in sharp relief about 2 mm. apart on the screen. A 4 mm. 
objective was used, and about 150 grating lines and two 
reference-lines were in focus in the field of view. On the screen 
was drawn a scale of divisions of which the central 100 were 
emphasized. The length of each scale-division was equal to 
the distance apart of the lines in the projected image of the 
grating. Thus by slight adjustment of the screen to one 
side or the other, and SO securing coincidence between the 
lines of the scale and of the orating image, 100 (or so) lines 
could be counted merely at a elance. There was no difficulty, 
therefore, in noting the number of rulings (always near 100) 
between each successive pair of reference-lines, the gratiny 
being racked along each time on the stage of the microscope 
by a convenient amount. 

Fig. 1 (Pl. XV.) is a photograph with a magnification 
of 450, and will give an idea of the appearance of the edge 
of the grating together with two of the reference-lines. 

The method proved to be very expeditious. Without 
interruption 1000 lines could be counted in 4 minutes; in a 
24 hours sitting 22,000 lines were enumerated without 
fatigue. A good fraction of this time was taken up with 
such things as attention to the illuminating are of the 
microscope, renewal of carbons, refocussing, &e. 

It may perhaps not be without interest to add that in the 
case of Mr. Gifford’s grating, two independent countings 
and a supplementary check counting agreed in giving a total 
of 45,668 rulings. 

At 16°-0 ©. on the hydrogen scale, these rulings occupied 
a length of 8°03618 cms., as the result of a comparison 


Counting the Rulings of a Diffraction Grating. 717 


against the Standard Invar Metre of the National Physical 
Laboratory which has been repeatedly verified at Sévres. 
This is at the rate of 5,682°57 spacings to the em. or 14,433°7 
to the inch at 16°°0 C., which may be compared with 
Brashear’s certified value of 14,438 rulings to the inch, 
temperature unknown. 

For Mr. Gifford’s grating the mean spacing value over. 
the whole of the grating was determined, but there would of 
course be no ditticulty in obtaining its value over any parti- 
cular region should local variation be suspected. 

In most of' the accompanying photographs (taken wi ha 
magnification of 690, which is reduced in the reproductions 
to about 450) the er ystalline structure of the speculum metal 
shows up strongly. 

Fig. 1 (PI. XV. ) shows the remarkably straight edge fiomted 
by the he where the ruling diamond was ‘set down at the 
beginning of each stroke. The other and far more irregular 
edge where the diamond was lifted from the metal is seen in 
fic. 2.. .The two sides of each furrow in the metal and the 
remains of the ‘ ‘cuttings ” can be plainly seen at the ends of 
some of the longer rulings. It is evident that the rulings are 
not very light ones, and that the original flat surface of the 
speculum has been completely replaced by a succession of 
fully developed V-shaped furrows with no intervening plane 
surface. This renewal of the surface is further illustrated 
by fig. 3, which shows a corner of the grating. The scratch 
across the corner was,.as will be seen, alinbat entirely removed, 
by the rulings. We should expect with such a surface that 
the apparent width of the ruling would depend on the obli- 
quity of the illumination. Such variation from line to line 
in the width of the rulings is noticeable in some of the 
photographs. Fig. 4 4 shows a blemish on the erating. The 
diamond appears to have got fouled for a number of. strokes 
and has lacerated the surface in some rulings and failed to 
rie at all elsewhere. 

I wish to thank Mr. Gifford-for his permission to include 
in this paper the photographs and measurements of his 
grating, and I am indebted to Dr. Glazebrook for his interest 
in the work. 


The National Physical Laboratory, 
Teddington. . 


aie 


LXXX. The Expansion and Thermal Hysteresis of Fused 
Silica. By G. W. C. Kaye, B.A., D.Sc. The National 
Physical Laboratory *. 


USED silica or quartz glass has assumed such importance, 

and has been applied to so many purposes in physics 

and chemistry, that a study of its thermal expansion may 
perhaps be of general interest. 

As is well known, quartz, which in the crystalline state 
has a considerable coefficient of expansion, assumes when 
fused a smaller coefficient of expansion than that of any 
other known substance, good invar alone exceptedf. For 
example, at ordinary temperatures the expansion coefficients 
are 

quartz, Ai iaxis: dS hO5 Te 
99 a 99 13°7 a2. 
fused silica, 6. Oa 


Owing to the extreme smallness of the expansion coefhicient 
of fused silica, most observers have adopted modifications of 
Fizeau’s interference method, more especially when for some 
reason it was convenient to work with small samples. 


I, THE CokEFFICIENT OF EXPANSION. 


Moderate Temperatures. 
Chappuis and Scheel have each determined the coefticient 
of expansion at moderate temperatures. Chappuis§ (1903) 


for the range 0° to 83° C. obtained the expression 


= = (385t + 001152?) 10-8, 
0 


where J; is the length at ¢°, J) that at 0°. . 
Scheel || in 1903 derived for the range 0° to 100° C. the 


formula 


2 = (-322¢ + 0014742) 10-8. 


In 19079 Scheel repeated his measurements with a new 


* Communicated by Dr. R. T. Glazebrook, F.R.S. 

+ Invar is obtainable as such in three grades, covering a range of 
coefficients of from about —0°3 x 10—§ to +25 10-6, 

t Bendit, Trav. et Mém. du Bur. Intl. i. 1881; vi. 1888. Scheel, 
Ann. der Phys. ix. p. 837 (1902). Randall, Phys. Rev. xx. p. 10 (1905). 

§ Chappuis, Procés Verbaua, Inter. Comm. des Poids et Mesures, 1903, 

75 


‘4 | Scheel, Deut. Phys. Gesell. Verh. v. p. 119, March 1908. 
Scheel, chd. ix. p. 718, Dec. 1907; Zert. Inst. xviii. p. 107 (1908). 


Expansion and Thermal Hysteresis of Fused Silica. 719 
cylinder of silica prepared by Zeiss. His new formula reads 


3 = (“388t + -001682t?—-0;504#°) 10-8, 
0 


which gives results in good agreement with Chappuis’ over 
the same range. 

The results in Table I. give for the range 0° to 100° some 
values of (;—1)/lo, derived from the above formule. 
_ A result of Randall’s (see later) at 80° is also tabulated. 


TABLE I. 
(le ae Lo) [Eo 
Scheel. 
. (2). 
Temp. (¢) Chappuis. Randall. 
1910. 
1903. 1907 
x10~° x10” x107° «107° 
| 
10° © 4-0 3-4 40 
50 22 20 ze 
80 38 35 a - 
100 50* 47 50 


* Extrapolated. 


Low Temperatures. 


At low temperatures the experiments of Scheel and of 
Dorsey claim attention. In January 1907 Scheel ¢ pub- 
lished some results dealing with the expansion over the range 
—190° to +16° C. of a silica cylinder made by Heraeus. 
These were embodied in the formula 


"= (-2174-+-002380%) 10-5 
0 


which indicates a minimum length at —46°. 
Later in the year Scheelt gave the results of experiments 
on}a silica cylinder made by Zeiss. They are represented 


t+ Scheel, Deut. Phys. Gesell. Verh. ix, p. 3, Jan. 1907, 
} Scheel, zbed. ix. p. 718, Dec. 1907. 


720 Dr. G. W. C. Kaye on the Expansion and | 


over the range —190° to + 100° C. by the formula (also given. 
above) 
Hee (-388¢ + 0016820 —-0,50403) 10-8, 
0 

which indicates a minimum length at —84°, and that the. 
length at —157° is equal to the length at 0°. — 

Dorsey* in 1907, using Fizeau’s method, worked over a 
range of —170° to +10° C. with two samples of | silica, 
tubing, one transparent, the other not. The latter shows 
results which are out of sequence. The results for the trans- 
parent specimen are given by Dorsey as coettiicients of ex- 
pansion for isolated ranges of 20° round and about various 
temperatures selected. ‘To facilitate comparison these results 
have been graphically interpolated by the writer, and so 
caused to yield the expansions at a suitable number of equi- 
distant temperatures—from which by summation. the vaiues 
of (l,—bo)/lio, and thence of (1;—J,)/l,, were obtained by 
making use of Chappuis’ and Scheel’s results (above) for the 
range 0 to 10°. A selection of these final values thus 
obtained are tabulated (along with Scheel’s) in Table II. 


TaAgii 1a) 
(le—Z,)/0o. 
Temp. (¢). | Scheel, 1907. 
| Dorsey, 1907. 
| | ' (Heraeus.) (Zeiss. ) 
Lea x10° x10 ° 
* —10° C. —1°9 —3°7 —3'2 
—20 —3-4 —70 —64 
— 50 —49 —146 alee 
—80 . = Sat A TF, —14-7 
(—84) (+0°6) (~17-7) (—146) 
—100 +271 —14:9 —134 
—150 | +21 =34- + 40 


— 190 +47 +21°6 +17:2 (—170°) 


Their graph indicates a minimum length at —77° and gives 

l_ys5=l,, Dorsey’s non-transparent specimen also showed 

a minimum length at about —80°. Having regard to the 
* Dorsey, Phys: Rev. xxv. p.-88, July 1907. 


Thermal Hysteresis of Fused Silica. 721 


above method of reduction of Dorsey’s results, their agree- 
ment with Scheel’s corresponding values for his Zeiss 
specimen must be regarded as very satisfactory. A mean 
of Scheel’s Zeiss values and Dorsey’s reduced values for low 
temperatures is probably not far from the truth. 


Ligh Temperatures. 


Le Chatelier * in 1900 was the first to make systematic 
measurements on the expansion of fused silica at high tem- 
peratures. Subsequent work has not confirmed his results, 
which depend on the expansion of porcelain. When plotted, 
his readings indicate a maximum length at 750° and a mean 
expansion coefficient of *67 x 10—° for the range 0 to 1000° C. 

Callendar f in 1901 obtained the value -}9 x 10—° for the 
mean coefficient of expansion between room temperature and 
1000° C. of a silica rod 40 cms. long. He states that the 
expansion is uniform up to 1000° C., increases rapidly 
from 1000° to 1400°, and changes to a contraction beyond 
1400°¢. The length was measured by a micrometer 
microscope, and the temperature was estimated by the ex- 
pansion of a surrounding platinum cylinder which was used 
to heat the silica. Callendar’s result reduced to the interval 
0° to 1000° is noted in Table ITI. 

Holborn and Henning § in 1903 used a rod of silica 52 cms. 


_ long, and by a microscope method measured the length at 


room temperature, 250°, 500°, 750°, and 1000°. The rod 
was heated electrically in a porcelain tube, and the tempera- 
tures at different points of the rod were measured by a 
thermocouple. The results have been reduced and brought 
into line with those of observers in Table IIT. 

Minchin || in 1907 found *45x10-® as the uniform co- 
efficient of expansion between room temperature and 950°. 
Certain errors have, however, since been found in his work. 

Randall ¥ has recently completed a comprehensive series 
of measurements using Minchin’s specimen of silica. He 
employed an interference method over the range 16° to 
1100°. The silica (a ring about 10 mm. long made by Zeiss) 
was heated in vacuo by an electric furnace ; temperatures 
were taken by a Pt-Rh thermocouple. The experiments — 


* Le Chatelier, Compt. Rend. cxxx. p. 1703 (1900). 

+ Callendar, Chem. News, Ixxxiii. p. 151 (1901), 

{ Shenstone, ‘Nature,’ lxiv. p. 65 (1901), gives this temperature as 
1200°. 

§ Holborn and Henning, Ann. der Phys. x. p. 447 (1903). 

|| Minchin, Phys. Rey. xxiv. p. 1 (1997). 

4 Randall, Phys. Rev. xxx. p. 216 (1908). 


Phil. Mag. 8. 6. Vol. 20. No. 118. Oct. 1910. 3B 


(pee Dr. G. W, C. Kaye on the Hepansion and 


appear to have been conducted with great care, and the 
results are entitled to considerable weight. They have 
the advantage of not depending on any assumptions as to the 
effect of temperature and pressure on the refractive index of 
air at high temperatures *; such effects are of the same order 
as that due to the expansion of the silica. Randall’s final 
values, reduced as before to the range (¢°—0°) by the aid 
of Chappuis’ and Scheel’s values for the interval 0°—16° are 
given in Table IIT. 


TABLE III. 


| (1 —2,)/%. 
Temp. (¢). | aa. a 1 
| Randall, | Holborn & Henningt, Callendar, ° 
| ToL 7 1903. 1901. 
x10" | x10” x10° 
200° C. | 100 | 
ee, feat Mocs, / 118 
300 161 
400 222 | 
500 281 276 
600 | 336 | 
700 389 
yy oy ‘Veh Capen eee | 404 : 402 
800 | 434 
900 | 481 | 
1000 541 | Dart 587 
1160 641 | 


| ah 


In fig. 1 the expansion per unit length (l:—lo)/ly is. 
plotted against temperature (t). The results of Randall, 
Scheel (Zeiss specimen), Dorsey, Chappuis, Holborn and 
Henning, and Callendar are utilized. | 

The excellence with which the curve represents the dif- 
ferent results will be noticed. The values of Holborn and 
Henning, and of Randall at high temperatures are in close 
agreement, as are those of Scheel and Dorsey at low tempe- 
ratures, and of Scheel and Chappuis at moderate temperatures.. 
It would appear that the curve represents with a considerable: 
degree of accuracy the expansion of clear, transparent, an- 
nealed silica from —190° to +1100° C. The inset gives a 
portion of the expansion curve on a larger scale. 


* Pulfrich’s formula (Zed. Inst. xiii. p. 455, 1893) has not been tested. 
for temperatures above 100° C. 

+ Henning, Ann. der Phys. xxii. p. 638 (1907) quotes values which 
when reduced read 132, 267, 402, 537 x 10-6 for the temperatures given. 
in this column, 


Thermal Hysteresis of Fused Silica. 723 


Randall’s value at 1100° confirms Callendar’s statement as 
to the rapid increase in the expansion of silica at tempera- 
tures above 1000°. The dotted portion of the curve beyond 
1100° is qualitative only and is an expression of Callendar’s 


observations. 
] . 


Fig 


re 


700 x10~° 
LECHATELIER 
600 
500 10° 20° 
TEMP(2). Ss 
400 
THERMAL EXPANSION OF FUSED SILICA 

300 


© PANDALL,/9/0. 
© SCHEEL, /907 
© Dorsey, 1907. 
+ CHAPPUIS, 1903. 
@ HOLBORN 
AND 

HENNING, 1903 

@ CALLENDAR, /90/ * 


200 
| eet 
100 “ 


-200 \., 2 0 200° 400° 600° 800° 1000° 1200°C 


EXPANSION PER UNIT LENGTH, (d;-Z)/Zp 


TEMPERATURE (2). 


The slope of the tangent to the curve at any point gives, 
of course, a measure of the coefficient of expansion 


(o=3.e 
arith: Ti) 

at that temperature, and the curve shows that « is negative 
below — 80°, zero at —89°, slowly increases to a maximum 


at about 500°, diminishes somewhat up to about 900°, and 
afterwards rapidly increases. 


3B 2 


724 Dr. G. W. C. Kaye on the Expansion and 


From the curve, one may derive the mean coefficient of 
expansion over any desired range between —190° and 1100° C. 
The following values have been obtained from the original 
curve which has been greatly, reduced in fig. 1. 


Temp. range. Expansion coefficient a. 


| 
| 
Te | 


~160° to —120° C. ~-43x10-" 
—120 to — 80 cr NAN 
— 80 to — 40 | +14 ,, 
— 40 to 0 | 4 hs 
0 to 30 | gh 
sO. te +200 Oy |e 
100 to 500 | TES «+ ay 
500 to 900 | OU 
900 to 1100 ‘80 


These results refer to clear, transparent, annealed silica. 
It is probable that want of annealing is the cause in most 
eases of the anomalous results which have been obtained by 
some observers. This view is supported by the experiments 
carried out in this Laboratory on the behaviour of specimens 
subject to a first heating as contrasted with their behaviour 
on subsequent heatings. 

In regard to the expansion of the translucent or satin-like 
variety of silica, information is forthcoming from some ex- 
periments of Mr. A. Blackie at the National Physical Labo- 
ratory, who has recently measured the relative expansions of 
the two kinds of silica, the transparent and the translucent. 
He finds that for temperatures below about 500° C. the 
translucent variety expands slightly more than the transparent, 
while for temperatures above 500° C. the reverse is true. 
The difference, however, is very small, not more, for example, 
than 30 parts in a million at 800°. 


II. CHance Points. 


It would appear from the expansion curve of fused silica 
that it shows at least two change-points, one at about 1000° C. 
the other at about —80° C. The former result agrees with 
the conclusion of Day and Shepherd*, who showed that for 


* Day and Shepherd, Amer. Journ. Sci. xxviii. p. 1089 (1906). 


Thermal Hysteresis of Fused Silica. 725 


all temperatures above about 1000°, quartz and fused silica 
devitrify into crystalline tridymite*, which above this 
temperature is the stable phase. 

At —80° Scheel noted a maximum density as well as a 
minimum length. There is no analogue of this point in the 
expansion curve of crystalline quartz f. 

As to the existence of a third change-point in fused silica 
at about 500°, the expansion curve, it is true, indicates a 
maximum value of the coefficient of expansion at that tempe- 
rature. But the maximum is not very pronounced and the 
certain existence of a change-point can scarcely be inferred. 
It is useful to note Mallard and Le Chatelier’s t measurements 
on the expansion of quartz crystal at high temperatures. 
Their results when plotted give an expansion curve which 
steepens up rapidly § in the neighbourhood of 500° to a well 
marked maximum length at about 670°; in fact, the expansion 
curve of crystalline quartz at 500° is very much like that of 
fused quartz at 1100°. 

A recalescence point at about 500° is well marked with 
quartz crystal, but a-thermo-junction embedded in powdered 
fused silica does not, according to Rosenhain, support the 
same claim for fused quartz. 

Perhaps it would be right to infer that the maximum at 
500° in the expansion curve of fused silica is a residual 
effect, and that the change from crystalline to amorphous 
quartz, though apparently complete as judged by other 
tests ||, is not so complete as to avoid recognition by the very 
delicate means that Tizeau’s method affords. 

However, the point, such as it is, may be useful in setting 
a limit to the temperature that should be employed in anneal- 
ing a silica standard of length intended for use at ordinary 
temperatures. Moreover, as will be seen later, the thermal 
hysteresis exhibited by silica is much less for temperatures 
below about 500° than for higher temperatures. A propos of 
this, Blackie’s observation (above) as to the reversal at 500° 
of the relative expansibilities of the transparent and trans- 
lucent varieties of silica may also be noted. 


* The densities of the various varieties of silica are :—Quartz 2-66, 
tridymite 2°32, transparent fused silica 2°21, translucent fused silica 2-07. 
The optical constants and crystalline forms of quartz and tridymite are 
very similar. See Dana’s ‘System of Mineralogy.’ 

+ See Scheel, Deut. Phys. Gesell. Verh. ix. p. 3, Jan. 1907. 

t+ Mallard and Le Chatelier, Compt. Rend, eviil. p. 1046 (1889). 

§ See also Randall, Phys. Rev. xx. p. 10 (1905). 

|| Mr. Blackie has obtained some interesting results at the N. P. L. 
from a microscopic examination with polarized light. 


726 Dr. G. W. C. Kaye on the Expansion and 


III. Toermat Hysteresis. 

Having regard to the expansion curve of fused silica, we 
should expect that any thermal hysteresis it may exhibit 
would depend very considerably on the temperature treat- 
ment. Callendar* remarks that if the temperature be kept 
constant at any point above 1000°, silica continues slowly to 
expand ; and furthermore, after such an expansion it does 
not return to its original length on cooling, but remains 
slightly longer. Randallt has noticed that this gradual 
lengthening at constant temperature in the region of 1100° 
is accompanied by,anisotropic expansion, the worked surfaces 
of the silica ceasing to be plane. ‘The distortion is moreover 
permanent and remains even after cooling. These results 
are of course not surprising in view of the existence of a 
change-point at 1000° C. 

At low temperatures, Dorseyt noticed that for a range of 
about 60° on either side of —80U°, fused silica shows this 
peculiarity ; above —80°, when warmed it first contracts 
slightly and then expands; similarly when cooled, it first 
expands a trifle and afterwards contracts. For temperatures 
below —80° the converse of this is true. Dorsey could not 
trace the effect below about —140° or above about —20°, 
nor did he notice it in any other substance. 

With temperature treatment which is not extreme, one 
may infer from a review of the observations of Holborn and 
Henning, Minchin, and Randall (see above), that the residual 
length alteration after a temperature cycle would be very 
slight if the silica has been annealed and if the temperature 
has not exceeded say 400° or 500°. The existence of such 

hysteresis for moderate temperature ranges has been defi- 
- nitely established and measured at the National Physical 
Laboratory by Mr. L. F. Richardson. 

To fix one’s ideas quantitatively, thermal hysteresis may 
be defined as follows:—Let / be the original length of a 
specimen, which is subjected to a rise of temperature of ¢, 
and is maintained at that temperature for say a day or two. 
If, when it is cooled to the original temperature, its length 


(after half an hour or so) is found to be (1+ 62), then ie is 


adopted as a measure of the linear thermal hysteresis H. 


; ‘ AL galt 
Since the mean coefficient of expansion= res where Al 
* Callendar, Chem. News, Ixxxiii. p. 151 (1901). 
+ Randall, Phys. Rev. xxx. p. 216 (1910). 
{ Dorsey, Phys. Rev. xxv. p. 88, July 1907. 


Thermal Hysteresis of Fused Siliea. w20 


is the linear expansion fora rise of temperature of ¢°, H may 
be looked upon as the residual variation of the expansion 


coefficient. 


Specimens of annealed fused silica both clear and trans- 
lucent were obtained in the form of end-measure rods about 
45 cms. long. They were subjected for periods of from 1 to 


90 hours to various temperatures over a range of from about 


—190° to 400° C. For annealed specimens, H, as defined 


above, came out between 1 x 10-° and 5 x 10-9%, which is less 


than 1 per cent. of the expansion coefficient at ordinary 
temperatures. Usually H was negative, which means that 
after heating and then cooling to the original temperature, 
the contraction was greater than the preceding expansion. 


This is in accordance arith the observations of Minchin and 


of Randall, and is, of course, one of the characteristics of 
invar. 

As will be seen below, silica compares very favourably 
with the two common Jena thermometry glasses specially 
designed to show a small after-effect. To extend the com- 
parison, Guillaume’s figures for invar are added. 


‘| Substance. | Temp. Range. | Linear Hysteresis H. | Authority. 
Jena 16'"* ...| O° to 50° 42 x10 ° | Whiesen & Scheel. 
dena 59’ T2020 stays 5D 23 Ps ‘A ori | 
| Invar t Ry Ae Be He O's tog 2s —8l i Guillaume. 
Ma fast cai | O “to 250 — 162 a ie 
Based Silica::.| @ ee Ald mule aege ee WN Pode 
in gy. uch O° HS eg : : 
- i 0 to >1000 +values Callendar. 


=(34+016¢)10~-°. + H=(27—008)10-%. + H=—3-25¢x10-%. 


Thus in regard to linear after-effect, silica over the range 
0° to 400° has nothing to fear in comparison with either 
invar or Jena thermometry glasses subjected, as will be seen, 
to much less severe temperature conditions. There is prac- 
tically nothing to choose between the difterent kinds of fused 
silica ; the cheaper satin-like variety is as good as the more 
expensive clear transparent kind. 

In justice to invar, it ought to be added that Guillaume’s 
observations on after-effect exterided over months, while the 
measurements on silica were included in the space of a few 
hours. 


728 Sir G. Greenhill on Pendulum 


It will be seen that fused silica has qualities which commend 
it for use as a material for standards of length. A silica 
standard metre is on the point of completion at the National 
Physical Laboratory, and there is good reason to believe 
that its adoption will be attended with success. 


The National Physical Laboratory, 
Teddington, 


LXXXI. Pendulum Motion and Spherical Trigonometry. 
By G. GREENHILL *. 


i R. ROSH-INNES has developed the relation between 

the revolution of a pendulum in a plane and the pro- 
jection of the motion on a spherical surface, and he shows 
that the argument of the elliptic function required can be 
represented by an area on the sphere which grows uniformly 
with the time (Phil. Mag. June 1910). 

In a change to the polar reciprocal, the time will then be 
represented by a spherical are, as discussed here in § 9. 

1. Consider a circle AQD on the vertical diameter AD, 
and a particle Q circulating round it under gravity with 
velocity due to the depth KQ below a horizontal line HK ; 
the motion of Q will represent a pendulum making complete 
revolutions, like a bicycle-wheel on its ball-bearings, put out 
of balance by an iron bar in the spokes (fig. 1). 

The lettering and notation is that employed in my ‘ Elliptic 
Functions,’ fig. 13, where, with ADQ=4¢, | 


(1) _KQ=AE—AN=AE—AD sin’ $ 


Pp ee AD 
=AK(i—<«’ sin? ¢)=AE. A’¢, k= AB? 
aa ViKielct BF (ap PY 29 .KP 
| ip)? _ yg Ax 
AEG 2 wing Se oat ele 
Sioa. ae, a) =x. 2? 


ue dd nt.) cc) wh a 
(3) \ Tre eS g=ama n=) AG? 


so that n/m is the number of beats per second in small oscil- 
lation; and the elliptic argument wu grows uniformly with the 
time ¢, starting from the lowest point A. 

Draw the circle, centre E and radius EB, orthogonal to 


* Communicated by the Author. 


Fig. 1. 


the circle AQD, cutting AD in L, the limiting point or 
Landen point ; then 

(4) HD. HA] UL’, and 2G) HQ’ H 1’, 

(5) QL?=(HQ—HL)?=HQ?—2HQ.HL+HQ. HQ’ 
=(HQ+HQ'—2HL)HQ=(2HM—2HL)HQ 
=2LM HQO=2hC KO 
=2LC.AE.A’d=AL?. A’d. 


Fig. 2. 


2. Turn the vertical circle about the diameter AD through 
a right angle, so as to bring DB to DO in fig. 2, and project 


730 Sir G. Greenhill on Pendulum 


in Mr. Rose-Innes’s manner on a sphere, by lines drawn 
from the centre O; corresponding points on the sphere in 
fig. 4 and the plane AQD of fig. 1 may be represented 
without confusion by the same letter, in most cases. ‘Then 


(1) = a = ae = sin? AOD; 


so that AOD is the modular angle, denoted by c; and 
KO=EL, and OL bisects the angle AOD. 
Invert; with respect to O, making O@.OQ’=OD?’; the 
Fig. 3. 


vertical plane AQD inverts into a sphere on the diameter 
OD, and the circle AQD into another circle A’Q’'D in a 
plane perpendicular to OA, so that these circles are circular 
sections of the cone, vertex O and base AQD. 

Now AQ is perpendicular to the plane ODQ, so that the 
planes OAQ, ODQ are at right angles, and the angle AQD 
on the sphere in fig. 4 is a right angle. 

If DX is the perpendicular on the tangent at Q, QDKX= 
QDA=¢in figs. 1 and 4; so also in fig. 4, if AY is the per- 
pendicular from A on the tangent at Q, QAY=QAD=¢’, 
but ¢' is the angle Q’A'D in fig. 2, or ADQ' in fig. 1. 

3. In fig. 4, by Spherical Trigonometry, 

(1) sn AQ = sin AD sin ADQ = « sin ¢, cos AQ=Agd, 

(2). sin DQ = x sing’, cos DO\=— Age 

(3) .cos AQ cos DQ = cos AD = cose, Ad Ad’ =«', 

(4) ¢' =am(K—uw), with¢d =amu. 


| 
c 


Motion and Spherical 7 rigonometry. 731 
Fig. 4. 


With EJ the polar circle of A, and E'J of D, and from 
(3) § 1, when Q makes a small advance to g, and M, M’ to 


! 
m, mM, 


‘ne _ dp _ Ad'dp _ cos DQ 


for Ag: «ale COS ¢ af 
__ spherical area QM'm'g 
i COs ¢ j 
dd? | Agddycas 2G 
pon es Ag’ ¢° Gee ene de 
__ spherical area QMmgq 
mm COSC ‘ 
a TC 
(7) a area AK’M a K—y = ore Pee 
COs ¢ Cos ¢ 
_ area AE’ JDQA 
Oat TS ee eee . 


The point Q describes a sphero-conic, with EJ, H’J the 


cyclic ares, since 
(9) cos AQ cos DQ = sin QM sin QM’ = cose, 


(Salmon, ‘Solid Geometry,’ Chap. X.), and the tangent 
UQU' intercepted by the cyclic arcs is bisected at Q, and 


732 Sir G. Greenhill on Pendulum 


cuts off a constant area UJU'=7—2c; so that the angles 
JUU', JU'U are equal to the angle UJU’, and then 
J9=]=O00=QU, JM=MU, J ae 

Since AU is a quadrant and AYU aright angle, YU is 
a quadrant, and so also is XU’; and XU'’=YU, QX=QY. 

If DX cuts AM in W, the spherical triangles XQW, 
YQA are equal, and 

(10) DWQ=QAY=DAQ=—¢q",  DWeDA; 

(11) DW=DX+XW=DX+AY=DA=c. 

4. So much for the geometry of the sphero-conic AQD, as 
developed in Salmon’s ‘Solid Geometry’ and by Mr. Rose- 
Innes; returning to the vertical circle AQD, draw another 
interior circle agd, centre c, with the same limiting point 
L and radical axis HEK, cutting QLQ’ in g, q' ; then (fig. 3) 

ay fle. dig = Bred a, 

(2) Qy.Qq7' =(HQ— Hg) HQ—H¢’) 

= HQ?— (H¢+ Hq')HQ+ HQ. HQ’ 
= (HQ+ HQ'’—Hg—Hy’) HQ 
=2Mm.HQ=2Cc. KQ. 

If QT is the tangent to this inner circle, cutting the outer 
circle again in Q,, 

(3) OF = 20h), G20. nee 

QT? 1? 

(4) Jie er 


OL? ~ OL 7 OL? 
and LT bisects the angle QLQ, ; also IT=IL, if QQ, cuts. 
HE in I, giving a simple construction of the inner circle for 
a given QQ). 
As the tangent QTQ, moves round, cutting the outer circle 
at equal angies, 


(5) vel. of Qh ss Q,T ne K,Q, 
weLvoly) GT. ¢ IA)? 
and this is the ratio of the velocity under gravity of two 
particles, Q and Q,, describing the circle in the same manner, 
so that Q and Q, will remain simultaneous positions of the 


particles if QQ, is a tangent of the inner circle ; and putting 
ADQ,=y=am wu, then 


(6) uy —u=w, a constant. 


5. Draw gef in fig. 2 through ¢ perpendicular to the plane 
AQD, cutting OA in f, and OL in g; the circle, centre f and 


Motion and Spherical Trigonometry. 733 


radius fO will pass through a and d, since 
(1) Ha. Ed=EL?=E0’, 
and 
(2) aOL=EOL—EOa=ELO—EdO=dOL, 


so that OL bisects the angle aOd, and passes through g on 
the circle round Oad. 

Then gla, gaO are similar triangles, and gl. gO =ga’, so 
that Land O are inverse points with respect to the sphere, 
centre g and radius ga; thence LI: TO is a constant ratio, 

equal to La:aO, and similarly LQ : QO is the constant ratio 
~ LA: AO round the circle AQD, and OQ=OA . Ag. 

The inverse of the circle afd with respect to O is another 
circle a'T'd' parallel to A’Q’D; for L’'T’:OL'=LT: OT, a 
constant ratio, so that L/T’ is constant, L' being the point 
inverse to L. 

Conversely the inverse of a system of parallels of latitude 
on a sphere with respect to a point O onthe sphere is a 
system of dipolar circles in a plane, as the circles of latitude 
on the stereographic representation of a hemisphere. 

6. The line O’T’ from O', the centre of the sphere on the 
diameter OD, makes a constant angle, c', with O'L’, and the 
angle DO’T’ is double the angle DOT’; so that if the arc 
DT in fig. 5 in the representation on a sphere, centre QO, is 
produced to double length to V, OV will make a constant 
angle ¢’ with OA, which is parallel to O’L’, and the are 
aC 


Then in fig. 5, by Spherical Trigonometry, 
(1) cos AT. cos DT =4(cos AD+cos AV)=4(cosc+t cosc’), 
a constant ; so that T describes another sphero-conic, interior 


734 Sir G. Greenhill on Pendulum 


to that described by Q, and with the same cyclic ares, so 
that its tangent QQ, cuts off from the cyclic ares a triangle 
UJU’ of constant area, and UU" is bisected at T (Salmon, 
‘Solid Geometry,’ §§247, 248). 


But since, in figs. 2, 3, 


Bee ay 

TQi LQu OQ,’ 

OT bisects the angle QOQ,, and Tin fig. 5 is the midpoint of 
QQ,, so that (Salmon, § 252) QQ, cuts off a constant area 
from the outer sphero-conic. 

With constant wu,—uw=w, the area QMM,Q, i is constant, 
so that the spheric: al quadrilateral QMM,Q, is constant, gall 
this implies that the sum of the angles DQQ,;, DQ,Q or DQV 
is constant, and this is found in § 8 to be am(K —w) +47. 

As before in fig. 4, XU’ and YU are quadrants, 
QY=Q,X=p suppose, Q.Y= (Q’X =g suppose ; 


DA=VY, DXFAY=AV=DW—ce 


7. The angle ADX=¢+wW in fig. 5, as in fig. 3; 
QDX=y, Q,DX=¢. 


Similarly DAY=¢'+w’, QAY=¢d', QAY=wW. 


(2) 


Then in the spherical triangle AQV in fig. 5, 
(1) sinQV=sin DQ,;=« sin’, sin AQ=<« sin , 


snQV am AQ. so xsing._ sin aa 

(2) sin QAV —*= sinQVA ~ sin QVA sin AQV’ 
so that QVA=¢; this is seen also from the equality of the 
triangles QYV, Qarx, in which QY=Q,X; YV=Dkiee 
that QVY= Q:DX=¢; this is the equivalent of Mr. Rose- 
Innes’s theorem (III.). 

Also 

(3) sin AQV= 


sin c’ 


os sin AQV= sin MQV= — cos DQV, 


since DQM is a right angle; and so we put, as in (8) $ 8, 
(4) AQV=am(K+w), MQV=am (K—w), 
sinc’ =«sn (K+w)=«sn (K—w), 
cos c’=dn(K+w) =dn(K—w). 


Motion and Spherical Trigonometry. 735 


8. The addition formula of the elliptic function follows at 
once from a Legendre spherical triangle, in the manner 
employed by Mr. Kummell (‘ Analyst,’ 1878) ; for in the 
spherical triangle AQV, 


(1) 


(2) 


Ina 


(9) 


(6) 


(8) 


sin p= sin QY= sin AQ sin QAY=« sing sin W’, 
cos p= V(L—«’ sin’¢ sin*p’), 

cosAQ Ad 

cosQY  cosp’ 

sin AQ cosQAY _ «sin ¢ cos’ 


cos AY = 


sin AY = a Ci cos p ’ 
, | COV Ae 
COs YV _— cos QY ei cos p 
af * Xx . ! 
sin YV = 2 QV cos QVY _ «sin y'cos 


y] 


cos QY cos p 
cosc’ = cosAV= cos (AY + YV) 

_ AgAw'’—«’ sin ¢ cos ¢ sin W' cosy" 
oa 1—x’ sin’¢ sin?’ 


= dn(w+ K—w,) = dn(K—w) =dn(K+w). 


similar manner, with Spherical Trigonometry, 
sin QAY cosAQ _ sin p’Ad 
cos QY te tear 32 
cosQAY _ cosy’ 
cosQY ~— cos p’ 
sinQVYcos QV _ singAy’ 
cos QY ia con po 
cosQVY _ cos 
cosQY - cosw’ 
cos AQV = cos (AQY + VQY) 
__ sin Ad sin Aw’ — cos ¢ cos ¥°’ 
i 1—x’? sin*d sin?’ 
= —cn(u+ K—u,) = —en(K—w) =cn{K +-w) 
AQV=am(K+w), DQV=t7r+am(K—w),. 


cos AQY = 


sia AQY = 


cos VQY= 


sn, VOY = 


9. In the reciprocal diagram of fig. 4, drawn on the left 
hand, R is the pole of XY, and the perimeter of the triangle 


7 


736 Sir G. Greenhill on Pendulum 


ARD is 7; the tangent intercept ZZ' by the cyclic ares 
is 47, so that QZZ’' is a spherical triangular octant ; 
RZ=ZK, RD=DK, RA=AK’; and if R, Z movestovene 


as Q advances to g, and Dz crosses ZR in 2’, 
(1) cose du = cos DQdd= sin DZdd = Ze’ 
= are Rr—rz+ RZ, 
(2) uweose =arcHR—RZ, 


thus representing the time by the difference of the arc HR 
and RZ. 

# In the reciprocal part of the diagram in fig, 5, where the 
tangents at R, R, intersect in §, 


(3) AS+DS=7—-—c, 

(4) (w—u) cose= are RR;— RZ, + RZ; 

(5) arc RR, =SR+S5SR,— constant (Salmon, § 252), 
(6) (uy—u)cose+aconstant=SZ—SZ,=8Z+48Z,'—4. 


The Spherical Trigonometry interpretation is the same as 
before in § 8; since 


(1) g=XQ=X8Q=37— DSZ=47—ASZ,/’, - 
sin g = sin DQsin QDX=« sin ¢' sin yp, 


é _cosZDS cosy 

OOS aa emg ann 

sin ZDSsinDZ __ sinyAd’ 
sin DSL hi 0) cong 


sin SZ = 


+ pe cos Z'AS __ cos op’ 
ep) eA sin ASZ,'~ cosq’ 
in SZ! = Sn ZtAS sin AZ _ sin p'Ay 
ROM) Fir Sie REIS TOME O™ cos AU 
(10) in OT4- 87,7) H Snob Ae +8 Se 


Cos” g 
= sn (K—u+u,)=sn (K+w), 
SZ+8Z,/=am (K+ w). 


Motion and Spherical Trigonometry. 737 
And with 


(11) p=XQ,=X8Q,=47—DSZ,=}r—ASZ’, 


cosSAZ! _ cosy’ 
sin ASZ' cosp ’ 


(2). sin SZ = cos Sh 


_sinSAZ’sin AZ’ sin Ad 
atte sin ASZ! re Wika a 


sel Dd 
sin DSZ, ~ cos p’ 
sin SDZ, sin DZ, _ sin paw’ 


(13)... cosSZ,= 


sin SZ,= 


sin DSZ, cas p ’ 
ve / / 
a) ain (SZ—84,) 2 — P cos Be p sin y’AdAy 


COS” Pp 
=cen(u+K—u,) = en(K—w), 


(15) SZ—SZ,=t7r-—am(K—w), 
LZ, Z,'=37= am (K+w) + am (K—w) —37, 
(16) SZ+8Z,'= am (K+). 


Thus the constant in (6) is am(K+w)—47—«'w. 


The sphero-conic ERE’ of fig. 4 is the projection on the 
sphere of the polar reciprocal of the circle AQD of fig. 1 
with respect to D, and this is the parabola ER; while the 
sphero-conic $8 in fig. 5 will be the projection of the 
hyperbola of S in fig. 2 polar reciprocal of the circle aTd. 

10. The motion of P at the same level as Q in fig. 1, 
oscillating on the arc BAB’ of the circle on the diameter 
Ali, will represent the associated motion of a pendulum, 
swinging through an angle 4c, and then if @ denotes the 
inclination of the pendulum OP to the vertical 


AN _AD AN 
AE” AE’ AD 
cos 30=Ad=cos DQ, 40=DQ, in fig. 4. 


(1) sin?l@= 


= x’ sin’ ¢, 


Draw a circle through B and B’, centre 0, in fig. 6; draw 
PB, and PpB' crossing this circle at p; and draw PH and 
Og perpendicular to BB! and PB’. 

Phil. Mag. 8. 6. Vol. 20. No. 118. Oct. 1910. 30 


738 Sir G. Greenhill on Pendulum 


The triangles PpB and OoB are similar, and so are the 
triangles PHB, PQ; so that 


(2) fm Wes ee sega 
F Oe" 0B PO. Per eB,’ 


and if PRP, is tangent to this circle, 


(3) PE=Py, PB’ =200, PE). iP, Rh’? = 200 Fee 


If P and P, are particles oscillating with the velocity due 
to the level BB’, 


, vel.ofP /PH PR 
(4) LEP ON Ea Pe 


so that, as before in §4, PP, will continue to touch the 
circle, centre o, during the subsequent motion; and the 
centre o has been chosen so as to bring P, to the same level 
as Q,, and then PQ, P,Q, will continue to be horizontal, by a 
suitable arrangement of gravity. 

11. It is desirable to have a geometrical interpretation of 
the addition formula of (4) and (7), §8, on the plane in 
figs. 3 and 6; this is seen by drawing AT, AR to meet DX 
and EX, produced, drawn perpendicular to QQ, and PP, in 
W and W;,. 


Motion and Spherical Trigonometry. 739 
Then if P, and Q; come to Py and Q, when P and Q are 
starting together from A, when 
u=0, u,=w, and ADQ,»=y=amw, cos AEP,=Ay, 


DW - Ta KD 
(1) DA = : == 28 = cos ADQ)>= cosy, DW=DQ,; 


EW a oR, EP 
2 Lee ree we 5 We geese ge ae, Ps ae 
(2) EA — y= ok aah 8 cos AEP, Ay, EW, EP). 
(3) AQ=ADsing, DQ=AD cos 9d, 
AP=AE.x«csin@d=ABsingd, HP=ALE. Ad, 


so that, by Euclid VI. C, 
(4) DX= se =AD cos ¢ cos vp, 


av AQ.AQ AQ, =AD sin ¢ sin yy, 


(5) AY,= Ake =AH.«’*sngsiny 
=AD sing sin p=AY, 
Ee 
BEX,= 2a =AE .AdAy. 


: DW DX XW_DX AY xW 
8 1 eh ae eas a a AY 
RW AP ee Berea QoL = Ay 


SSS SS Se 
— —— —— a 


AY, EY" +e a 2 
(7) cosy =cos ¢ cosy sin > sin pAy. 
_ HWY BX POW A) AY ee 
(3) 4y=an'= an + An = AE + AR” “AY,” 
(9) XW, X,;R Ho  PoRo | DNo 
AVG” 2 BY wAaT. BigAy DA 
(10) Ay=AgAy+x’ sin ¢ sin cosy. 
Here (7) and (10) are the well-known formulas of Legendre; 
and thence, as before in (7) and (4), § 8, 
(11) cosy=cos¢ cosp+sin d sin p(AdAy+ x’ sin ¢ sin cos ) 
cos @ cosyr+sin gd sin pAdAy 
tee ar a aa, 
(12) Ay=AdgAv+x’sin p siny(cos ¢ cos ~+sin d sin pAy), 
A AdAv +x’ sind sin cos ¢ cos v 
‘ie 1—«’ sin’ ¢ sin’? p . 


== COS ¥y, - 


3 C 2 


740 Prof. E. G. Coker on the, 
Or, geometrically, 
cosy —cosp cosy AW EM a ae 
Stich 6m Pi!" A A ie eee ame i 
= AdAw +x’ sing sin cos y, 
(i, eet oe |. a Nagel ee 
« sindsiny AW, AD” AD 7 Alpe 
= cos cosy + sin d sin pAy, 


equivalent to (11) and (12) above. 
So also the formula for sin y=sn w can be interpreted. 


(13) 


LXXXII. The Optical Determination of Stress. By KB. G. 
CokER, J.A., D.Sc., Professor of Mechanical Engineering, 
City and Guilds of London Technical College, Finsbury *. 


HE principal advances in our experimental knowledge 
of the strength and properties of materials have been 
made by the use of mechanical apparatus for applying stress 
and measuring strain, and instruments of this class are now 
in general use possessing a sufficient degree of accuracy for 
the most refined measurements, but whatever applications 
purely mechanical methods may have, they possess a common 
characteristic feature that measurements must be taken over 
a definite length, area, or volume maintained in a standard 
condition throughout in order that the state of stress or strain 
may be referred to some standard measure possessed by the 
instrument, or by which it is calibrated. Whatever the 
arrangement may be, it is not in general possible to measure 
the stress or strain at a point, it the body is subjected to 
stress varying from point to point. 

This defect in purely mechanical devices is one which from 
the nature of the case is hardly likely to be overcome entirely, 
yet in the great majority of the problems which arise in 
practice the stresses change very rapidly from point to 
point, and experimental information, if it exists, has almost 
invariably been obtained by using mechanical apparatus 
incapable of determining the stress at a point. Mathematical 
researches of the state of stress and strain in bodies give 
exact solutions of a variety of complicated problems; but 
some of the simplest forms of practical construction offer 
problems of the greatest difficulty, as for example the deter- 
mination of the stresses in hooks, chain links, and rivetted 


* Communicated by the Author: read in abstract at the British 
Association, Sheffield. 


Optical Determination of Stress. 741 


plates, the effect of notches and holes of various forms in 
tension and compression members, beams, pillars, and shafts, 
the distribution of stress in built up structures such as plate 
girders, rivetted frames, masonry dams, and the like. 

In most of these cases the distribution of stress has not 
been completely solved. 

One of the most suggestive and instructive experimental 
methods is suggested by the differential equations of plane 
strain which, under certain conditions, have identically the 
same forms as the equations of stream-line motion in a 
perfect fluid. Thus in irrotational plane strain when wu and 
v are the displacements in the direction of the axes of « and 
y, we obtain 

dreldr—du/dy=0, Odu/dz+dr/dy=0, 
while if w and v are the displacements of a perfect fluid 
moving irrotationally we obtain similar equations. The 
stress problem has therefore an analogous problem in hydro- 
dynamics, but the restrictions to which the analogy is subjected 
seriously limit its applications. 

Many experimental researches on the behaviour of materials 
have also been made using models shaped in rubber, and by 
measurements of the comparatively large strains produced 
in this material under various conditions of loading, the 
stresses in the structure have been determined. 

A method invented by Brewster depends on the applica- 
tion of polarized light for observing the condition of a 
specimen made of glass, and he suggested that models of 
arches might be made of this material and their optical 
properties examined under stress. 

This matter has received a considerable degree of develop- 
ment at the hands of Neumann* and Maxwellt; but the 
difficulty and expense of making objects in glass has hindered 
the progress of the experimental method. 

In an attempt to obtain an optical verification of the 
mathematical theory of the stresses at the principal section 
of a hook, one or two models were shaped from a square of 
glass, and their behaviour under stress examined by polarized 
light. The experiments showed that to produce any measurable 
effect the glass must be very thick and the stresses dangerously 
near the breaking stress of the material. 

Several other materials were tried, and a suitable material 
was ultimately found in “xylonite,” which possesses most of 


* “Die Gesetze der Doppeibrechung des Lichts,”’ F. E. Neumann, 
Abhandlungen der K. Akademie der Wissenschaften zu Berlin, 1841. 

+ “On the Equilibrium of Elastic Solids,” J. C. Maxwell, Collected 
Papers. 


742 Prof. E. G. Coker on the 


the desirable features of glass, with the additional advantage 
of being easily cut into shape without suffering any injury 
of its elastic or optical properties. 

Xylonite is a preparation of nitrocellulose of widely ex- 
tended use, and it can be obtained in sheets several square 
feet in area free from initial stress except at the edges, so 
that it shows no colour effect in polarized light. 

It is not quite so transparent us glass, and it is usually 
slightly tinted, but this is not found to be a disadvantage in 
practice. 

Specimens may be cut from the sheet by ordinary wood 
and metal working tools, and with a little practice any object 
capable of being represented in a plane may be fashioned 
with ease. ‘he material is fairly isotropic in character, and 
although a sheet appears to show a distinct grain, this is 
apparently a mere surface effect due to the sheet being cut 
from a plastic slab ; it is afterwards subjected to treatment 
which renders it semielastic, and its surface is finally polished 
before use. 

Xylonite is much more compressible than glass, and it 
possesses the useful property of not readily breaking under 
stress. Up toa stress of about 4000 Ibs. per square inch it 
may be subjected to repeated stresses without injury or 
change of its optical character, but beyond this it shows 
signs of residual stress when the load is removed. 

Its elastic behaviour under stress is not so perfect as that 
of glass, and the values of Young’s modulus and Poisson’s 
ratio are very different. 

This will be readily seen when the results of experiments 
on specimens of both materials are compared. 

The following table gives the values obtained in a tension 
specimen of xylonite 8 ins. long between measuring-points, 


0°49 in. wide and 0°123 in. thick. 


TaB_eE I. 
| 
: Contraction 
Load, Extensions, apr cs t ian 
pounds. ina, Differences, | 0° oe 
| ins. 
LS Tee OE ere 0 a 
40 O17 ON coon 
80 035 ogy | 00085 agg 
LL I fae O71. ugg 0017 = .qg3 
ae Oe ule ee ae Oe 
40 ae 00055 — Wv045 


0 001 019 -0001 00045 


Optical Determination of Stress. 743 


These readings give a mean value for Young’s modulus 
of 299,100 in lbs. and inch units, and for Poisson’s ratio a 
value 0°39. 

For comparative purposes a set of similar values are given 
in Table II. for a piece of plate-glass which when examined 
between crossed nicols showed very little trace of internal 
stress, and in this respect it was very similar in character to 
xylonite. 

The specimen* was 1:016 ins. by 1°008 ins. in section and 
1°25 ins. of measured length. 


TasxeE II. 

Load, ) Compressions in | Lateral extensions, 
pounds. millionths of an inch. | millionths of an inch. 
3000 | 949 240 we 2 
= re 240 2 «644 

5000 | 480 5 40 86 48 

7000 . 720 930 134 44 

9000 | 940 220 178 43 

7000 Fiabe er | 
5000 480 240 89 49 

3000 | 240. 540 40 46 | 
1000 ) 0 | 


and the mean value of E obtained from these readings was 
10,380,000 and for Poisson’s ratio 0°233. 

If a homogeneous beam of plane-polarized light passes 
through a plate of unstressed glass, xylonite, or other like 
transparent material, it suffers no decomposition; but the 
application of a tension or compression stress causes the 
material to behave like a double refracting substance, and 
the plane-polarized beam breaks up into plane-polarized rays 
having their directions of vibration parallel and perpendicular 
to the axes of principal stress. These rays have different 
velocities in the material, and their relative retardation R is 
proportional to the indices of refraction fy and pe of the two 
rays and the thickness T of the plate of material through 
which they pass; this is expressed by 


R=(fp—pe)T. Le TE Gee a a (1) 


Hixperiments on glass show that the difference of the re- 
fractive indices is proportional to the difference of the principal 


* Specimen “d,” p. 65, of “An Investigation into the Elastic Constants 
of Rocks,” by Adams and Coker, Proceedings of the Carnegie Institution, 
Washington. 


ee ad 


744 Prof. E. G. Coker on the 
stresses X and Y in the plate, and if therefore we write 
fo—pe=C(XK-Y), . . 00 


where C is an optical coefficient, the value of which can be 
determined by experiment, we obtain 


R=C(X—Y)T. 


a a 


Wertheim’s* experiments showed that the optical co- 
efficient is independent of the wave-length of the light 
used, but the later experiments of Pockelst and Filont 
show that a variation exists which in very accurate experi- 
ments must be taken into account. For the purposes of this 
paper the variation of the optical coefficient is neglected, and 
the retardation is assumed to follow the law stated by 
equation 2. 

A convenient arrangement for examining the effects of 
stress is shown by fig. 1. Light from a point source A is 


Fic. 1, 


A 

| Ei 
Ft 
ae 


' 


tr 


ee 


Y 


| 
| 
| 


y- 


_ plane-polarized by its passage through a Nicol’s prism B, and 
is transmitted through a transparent plate C of the material 
under examination. This transmitted light is analysed by a 
second Nicol’s prism D, and the image showing the colours 
produced by the interference of the ordinary and extra- 
ordinary rays is projected on a sheet E of squared paper, or 
is photographed as ‘may be convenient. Condensing lenses 
F and G are also provided for focussing purposes. A very 


* Annales de Chimie et de Physique, Series 3, vol. xl. 


+ “Uber die Anderung des optischen Verhaltens verschiedener Glaser 
durch elastische Deformation,” Annalen der Physik, Series 4, vol. vii. 
¢ Camb. Phil. Soc. Proe. vols. xi. and xii. 


Optical Determination of Stress. 745 


convenient form of apparatus for this work is described 
by Cheshire*, and his arrangement was used for the 
experiments. 

If the value of the optical coefficient is known, and the 
retardation is measured for a given thickness of the material, 
the difference between the principal stresses at a point may 
be obtained, or in the case of simple tension or compression 
where one of the principal stresses is zero the absolute value 
of the stress can be directly determined. 

For most purposes, however, it is more convenient to pro- 
ceed in a different manner by a process of comparison of 
colours. A scale may be readily formed in which the relation 
between say tension or compression and colour due to inter- 
ference is obtained experimentally. Thus, for example, a 
specimen 0°49 in. wide and 0°123 in. thick gave the following 
colour-scale in tension. 


Tase II]. 
| 
Load in pounds eee Load in pounds | Gainer 
per sq. inch. || er 8a. inch. | 
i Dark field. | 2020 | Reddish yellow. 
340 Faint white. | 2170 Reddish purple. 
670 Intense white. | 2350 Purple. 
1018 Faint yellow. | 2510 Sky-blue. 
1340 Lemon-yellow. || 2690 Very light blue. 
1680 Orange-yellow. 3020 Nearly white. 


and these loadings were repeated at different times with the 
same results as far as could be judged. 

It is difficult, however, to independently estimate most of 
these colours, with the possible exception of purple, which is 
usually well defined; and it is necessary for accuracy to 
arrange the experiment, so that the colours produced in the 
object under examination may be directly compared with a 
colour produced by uniform stress. 

It is not difficult to arrange a standard specimen and the 
object under test so that they may appear close together in 
the same field of view, and this allows of much greater 
accuracy as the colours produced may be matched in the 
same manner as in photometric work. 


* “Some new Optical Projection Apparatus,” by F. J. Cheshire. 
Optical Society, 1908. 


746 Prof. E. G. Coker on the 


The usual arrangement adopted is shown in fig. 2, in 
which a symmetrically loaded tension specimen A and another 
B with an eccentric load are both shown secured in the grips 


of miniature testing machines. In this example the specimens 
were cut from the same strip, and the interference effects 
were projected on to a squared paper screen so that the 
breadth of the eccentrically loaded specimen was 2 ins., cor- 
responding to an actual breadth of 0°309 in. 

The stresses produced at different points in the cross- 
sections were determined by loading the standard test-piece 
until the uniform colour produced in it agreed with that 
produced at a definite point in the cross-section of speci- 
men B, and the stress was calculated from the load and 
the dimensions of the levers. This method requires no cor- 
rection for the tension side of the specimen as the diminution 
of thickness due to the load is the same for both, but a small 
correction is required for the compression side as the relative 
retardation between the interfering rays is greater on account 
of the increase in thickness due to the stress. 

Measuring in this way, the following values (Table IV.) of the 
stress were obtained across the section of the specimen B, where 
the abscissee refer to the projected dimensions measured from 
the line of application of the load, which in this case coincided 
with one edge of the specimen. 

These values are plotted in fig. 3, and the diagram shows 
that the variation of stress in the specimen is approximately 
a linear one, except for the highest load, when the specimen 
failed on the tension side, thereby producing a change in the 


Optical Determination of Stress. TAT 
TaBeE IV. 


| Abscissz, inches. 
| 
| | Distance of 


(1 ee eet ys 0:5. 0-75. 4  neutralaxis | 2-00. 
| from zero line 
ins, 

oa | +1750 se +1190 pa | 1°35 | —960 
ite 2820 (iy veel) 41410 |e ce) | 1:35 —107 
£3” | +2880 | +2540 | +1880]... | 1:39 —1190 
2 ;, | +3330 | +2880 ee +1520 1:45 | —1470 
~—~ Oo / 

m2 | +3560 +3450 | +3160 | +2200 1-47 | ~1640 


/ych 
& 
S 
S 


+ 


STRESS FOUNOS PER SQUARE 


properties of the material, which showed itself when the load 
was removed by its faint doubly refracting power. The 
highest values of the stress are therefore not correct, owing 
to their values having been obtained by comparison with a 
test piece differing in optical condition from the object. 

The position of the neutral axis for the lower loadings 
agrees very well with that given by calculations using the 
ordinary formula for beams. It is also worthy of note that 
the neutral axis moved away from the tension side as the 
load increased—a result* which agrees with theoretical 
determinations. 


* Love’s ‘ Theory of Elasticity,’ p. 349. 


—_— — ewe eee ee ee 


EE 2 Se ee - eee SS SE EE Ee ee ee ee es ees we. 


| q 
i 
i 


748 Prof. E. G. Coker on the 


As an example of a more complicated kind we may take a 
hook of very great curvature. This case differs optically 
from the preceding one in a very important way, as the 
principal stresses on each side of the principal section show 
considerable variation in angular position. 

If, therefore, a plane-polarized beam passes perpendicularly 
through the plate with the intersection of its plane of polari- 
zation oblique to the directions of principal stress, it is 
resolved into components corresponding to these latter 
directions, and therefore in all parts of the specimen through 
which the plane-polarized light passes, the ray is resolved 
into two directions at right angles. The interference of 
these two rays produces colour fringes, except in those parts 
of the field where the directions coincide with the planes 
of the crossed nicols. Such bands, therefore, indicate the 
directions of the principal stresses, and by turning the nicols 
round while their planes of polarization remain at right 
angles to one another, a series of curves are obtained as 
shown, for example, by fig. 4, where these loci have been 


Fig. 4. 


obtained for a hook having an outer radius of 0°75 in. and 
an inner radius of 0°277 in., the plate being 0°123 in. 
thick. , 

The effect thus produced is equivalent to superposing a 
black cross upon the interference colours produced by the 
stress, and it serves a useful purpose in that it enables curves 
of principal stress to be drawn graphically, or by calculation 


ice tt eee 


a a 


Optical Determination of Stress. 749 


when these loci are determined with reference to the angular 
positions of the nicols. Snch curves have been determined 
graphically in this case and are shown by fig. 5. 


\ 
\ 
XN 
+A 
oN 
\ 
\ 


NCCC NEES NGG 
RRND XN \ aN \ : me 
NW \ 2 Ae 


For measurement of the stress it is necessary to get rid 
of the distorting effect produced by the black cross, and I am 
greatly indebted to Professor Silvanus Thompson, F.R.S., 
for suggesting for this purpose the use of two quarter wave- 
plates (H, I, fig. 1) set with their axes inclined at 45° to the 
Nicol’s prisms, whereby the plane wave issuing from the first 


‘nicol B is converted into circularly polarized light. The 
circularly polarized beam produced by this combination, | 


whether right-handed or left-handed, has no special direction 
of polarization, and it therefore presents the same aspect to 
all parts of the object under stress. It is again converted 
to plane-polarized light by the inverse combination of quarter 
wave-plate and nicol and the interference fringes are still 
produced, while the black cross disappears. This arrange- 
ment has the further practical advantage that, except for a 
slight and invariable change of tint, the interference colours 
produced are independent of the angular position of the 
object. 

Fig. 6 shows the general arrangement of the colour fringes 
presented by the hook when viewed by circularly polarized 
light, and the stresses at points of the central section can be 
determined in a similar manner to that described above. 


750 - Prof. HE. G. Coker on the 


The projected image of 0°473 in. actual width measured 
2 ins. on the squared paper screen, and the stresses obtained 
are given in the annexed table. 


WA = NEUTRAL AXIS 
LB= LIGHT browv 
B= BLUE 

P= PURPLE 


| Absciss, inches. 


2 - 0. | 016 0:25. o5, | Neural | | oe 
| Ee ees | ae ee SS ee S| ee See Seen eee nee | Wor er 
2:5 | 9380 1970 at 0-1 s 0-7 Ms 640 
; 3 3370 | 2380 at 0-16 el OT 950 | 1270 
22 | 4060 | 2480 at 0-25 Jee sn 1460 | 1720 
ay UAT (acct: cea wen 1460 | 0-75 | 1910 | 2730 


and a plot of these values is shown in fig. 7, 

The position of the neutral axis has been mathematically 
determined by Andrews and Pearson*, and its distance 7 
from the centre of the section has been shown to be 


ok dln oP enaoa ae 
Yo= Po "1 5 koh i 


* Drapers’ Company Research Memoirs. Technical Series. I. 1904, 


, 
q 
, 
d 
i 


Optical Determination of Stress. 751 


where py is the mean radius of curvature, ¢ is the distance of 
the applied force from the centre of the section, and 9, y2 
are constants depending on pp, 7,and T. In the present case 
y1=1:048, y2="891, where »='39, and the position of the 
neutral axis is ‘055 in. from the centre on the tension side. 


Fig. 7. 


TEwsioN Founos FER SQUARE lYCH 


COMPRESSION £88 PER SQN 


This agrees fairly well with the observed value of :059 in. 
having regard to the fact that po is increased about 10 per 
cent. of its original value by the load. 

Owing to the uncertainty of the exact position of the 
forces when a ring of such great curvature is stressed and 
somewhat distorted by a load, it has not been possible with 
the apparatus at my present disposal to establish any very 
accurate relationship between the external and internal 
forces. 

The present examples, however, serve to illustrate the 
practical uses which it is possible to make of this method of 
analysing the stresses produced in any object capable of 
being represented in a plane by a model cut from a sheet of 
transparent material. 


LXXXIII. Rays of Positive Electricity. 
By Sir J. J. Taomson * 


FIND that the investigation of the Positive Rays or 
Canalstrahlen is made much easier by using very large 
vessels for the discharge-tube in which the rays are produced. 
With large vessels the dark space around the cathode has 
plenty of room to expand before it reaches the walls of the 
tube ; the pressure may therefore be reduced to very low 
values before this takes place, and in consequence the potential 
difference required to force the discharge through the tube 
at these low pressures is much lower than when the tubes are 
smaller. It is possible with large tubes to work with much 
lower pressures than with small ones, and at the lower pres- 
sures phases of the phenomena of the positive rays come to 
light which are absent or inconspicuous at higher pressures. 
With small tubes and therefore comparatively high pressures, 
when the arrangement used to investigate the | rays is that 
described in my former paper (Phil. Mag. [6] xvi. p. 821, 
1909), 2. e. when the rays passing from a ‘hole in the cathode 
through a long narrow tube fall on a phosphorescent willemite 
screen after passing through superposed magnetic and electric 
fields, the appearance on the screen is as follows. 


The bright spot A which marks the place where the 
undeflected rays strike the screen is drawn out by the magnetic 
and electric forces, producing respectively vertical and hori- 
zontal displacements, into a straight band AB (fig. 1) of 


* Communicated by the Author. Read at the meeting of the British 
Association, Sept. 1, 1910. 


Rays of Positive Electricity. 733 


fairly uniform intensity ; there is also a fainter prolongation 
AC of the band in the opposite direction to AB due to rays 
which carry a negative charge. The velocities and the values 
of e/m for the rays can be determined by measurements of 
this band. For if y and w are the vertical and horizontal 
deflexions of a ray striking the screen at P, then the velocity 
of this ray is equal to cy/« and the value of elm to cay, 
where ¢, and ¢, are constants depending on the strengths and 
positions of the electric and magnetic fields. I have shown 
(Phil. Mag. loc. cit.) that the velocity of the rays in this 
case is practically independent of the potential difference 
between the electrodes in the discharge-tube, and that we 
could increase the potential difference from 3000 to 40,000 
volts without appreciably increasing this velocity. With 
small tubes the appearance I have just described is often the 
only effect to be observed even when the pressure is reduced 
close to the point at which it ceases to be possible to force 
the discharge through the tube. 

When large discharge-tubes are used a much greater 
variety of effects can be observed. I have used tubes with a 
volume as large as 11 litres; these, however, are somewhat 
difficult to procure and not very convenient to work with. I 
have found flasks having a volume of 2 litres, such as are used 
for boiling-point determinations, large enough for most 
purposes. 

A uniform and sensitive phosphorescent screen is of great 
importance as there is often a considerable amount of detail 
to be made out, and some of it too faint to be visible unless 
the screen is a very good one. My assistant Mr. Everett 
has lately succeeded in making very uniform screens by 
grinding the willemite into exceedingly fine powder, then 
shaking the powder up in alcohol and allowing it to settle 
slowly from the alcohol on to a flat glass plate ; when the 
deposit has reached the requisite thickness the rest of the sus- 
pension is drawn off and the deposit allowed to dry ; when 
dry it sticks quite firmly to the plate, and the deposit is much 
more uniform than that obtained by the method I formerly 
used of dusting powdered willemite on a glass plate smeared 
with water glass. The screens soon lose their sensitiveness 
if bombarded by the rays, and when any fine detail has to be 
made out it is advisable to use a new screen or a part of the 
screen not previously bombarded by the rays. | 

The discharge-tube is shown in section in fig. 2 (p. 754). 
The perforated cathode C protrudes well into the tube, the 
rays pass through the hole in the cathode through the fine tube 


Phil. Mag. 8. 6. Vol..20. No. 118. Oct. 1910. 3D 


154. Sir J.J. Thomson on 


and then travel between the poles MM of an electromagnet 
and the parallel plates PP which are connected with a battery 


Fig. 2. 


+ 


of small storage-cells ; the rays after being defected fall on 
the willemite screen 8S. Witha tube of this kind the appear- 
ance on the screen as the pressure is gradually reduced is as 
follows, the rays being exposed to both magnetic and electric 
forces. At the highest pressure at which the phosphorescence 
is visible, the phosphorescent patch covers a considerable 
area, the left hand (the least deflected) boundary being fairly 
well defined while the other boundary is hazy. As the 
pressure is still further reduced we get the appearance shown 
in fig. 1 ; this persists for a considerable range of pressure, 
but as the pressure is still further reduced bright spots as 
described in my paper (Phil. Mag. [6] xui. p. 561, 1907) 
begin to appear, while the luminosity appears to divide into 
two portions, the appearance being that represented in fig. 3. 


Fuel 8i 
L 


The luminous band, which at the higher pressures was the 
sole representative of the phosphorescent, can still be seen in 
its old position though it is not so bright as when the pressure 
was higher, the negative continuation of it still persists. As 
the pressure is still further diminished this part of the phos- 
phorescence with its negative accompaniment gets fainter 
anc fainter but does not alter in position, showing that the 


Rays of Positive Hlectricity. 759 


velocity of the rays producing it is independent of the 
potential difference between the electrodes, finally when the 
pressure is very low it looks like a faint nebulous band over 
which brighter patches are superposed, 

The relations between the positive and negative portions of 
the phosphorescent figures when the pressure is low are 
very interesting. The lower, more deflected portion has 
frequently two “bright spots A and B for each of which 

e/m=10!: one at A which gradually moves, as the pressure 
is diminished, along a parabolic path to O, the position of the 
undeflected spot; the other, not quite so definite, at B, a point 
on the phosphorescent band which has survived from the 
higher pressure. The negative portion at these low pressures 
is not a replica of the positive portion as it was at the higher 
pressures, but remains unaltered in shape and position as the 
pressure diminishes, getting gradually fainter. There is no 
trace on it of the spot A ; the spot Bis, however, visible at B’, 
and the luminous band BB’ can be traced as a ‘straight strip 
occupying the same position as it did at higher pressures 
when it was the only part of the phosphorescence visible. 

There is nothing on the negative side corresponding to the 
portion OCD on the positive, or at any rate if it exists it is 
so very much fainter, that I have never been able to satisfy 
myself of its existence, even when tiie negative part OB’ was 
quite bright. 

I think there is exceedin gly strong evidence to show that 
the straight band of phosphorescence which alone is seen at 
higher pressure and which lingers on with diminished intensity 
when the pressure is reduced, has a different origin from the 
phesphorescence which shows itself as bright spots on an 
isolated streak of phosphorescence, and which is due to rays 
whose velocity, unlike that of those producing the first kind 
of phosphorescence, depends upon the potential difference 
between the electrodes. 

Such evidence is afforded by the following experiments, 
the first of which shows the complete symmetry between the 
positive and negative parts of the first kind of phosphorescence, 
and also that much of this kind of phosphorescence is due to 
secondary rays produced after the primary rays have passed 
through the cathode. In this experiment, the magnetic and 
electric fields, instead of being as in the previous experiments 
arranged so that when a particle was exposed to a magnetic 
force it was simultaneously exposed to an electric one, were 
made to overlap. The poles MM of the electromagnet were 
pushed nearer the screen so that they extended on the screen 
side beyond the parallel plates PP which produced the electric 


“2 
ad 


756 Sir J. J. Thomson on 


field. With this arrangement a particle, after leaving the 
space between the plates, enters a region where it is exposed 
to magnetic but not to electric forces, 7. e. when it is deflected 
vertically but not horizontally. In this case the appearance 
presented by the phosphorescence patch at the pressure, when 
aa normal circumstances it would be as represented in 
fig. 1, is shown in fig. 4 

“hore is now a vertical portion OA due to 
rays which have been deflected vertically but 
not horizontally, i. e. which have been acted 
upon by magnetic but not by electric forces, 
and which must therefore have been produced 
between the ends of the parallel plates and 
the screen. Connected with the vertical piece 
OA there is a curved part AB due to rays 
which have been deflected by the electrostatic 
as well as the magnetic forces. The rays 
falling on the portion of AB near to A have 
been produced inside the parallel plates close 
to the end next the screen, and have only 
been exposed to the electric force for a small 
portion of their path. As we approach B 
the corresponding rays have been produced 
nearer the cathode, while the rays at the 
very end were already produced before the space between 
the plates was entered, for we find that the end B of the 
phosphorescent patch is in the same position as where the 
fields of action of the magnetic and electric forces were 
coincident. 

If we reverse both the electric and magnetic forces so as to 
bring the negatively charged particles on to the part of the 
screen previously occupied by the positive ones, we find that 
the phosphorescent band due to the negative rays is an exact 
reproduction in shape, size, and position of that due to the 
positively charged particles. 

If this experiment is repeated when the pressure has been 
reduced to the stage when the phosphorescence splits up into 
two bands as in fig 3, the contrast between the behaviour of 
the two bands is very instructive. The lower band (7. e. the 
one where the magnetic deflexions are the greatest) is bent 
in the way we have just described, and is below the position 
it occupied when the magnetic and electric fields were coin- 
cident. The upper band on the other hand is bent in the 
opposite way and is above the position it occupied when 
the fields coincided. The appearance of the phosphorescence 
is represented in fig. 5, where the dotted lines show the 


Fig. 4 


B 


sn > 


Rays of Positive Electricity. 757 


positions of the bands when the magnetic and electric fields 
coincide, the continuous lines their position when the magnetic 


Fig. 5. 


%& 


Ye, He) 
ie 


field is pushed forward towards the screen. The shape of the 
lower band can be explained as we have seen by supposing 
that it is due to secondary rays which are continually being 
produced as the undeflected rays travel from the cathode to 
the screen. The configuration of the upper band can be 
explained by supposing that it is due to primary rays coming 
through the cathode, and that these are not recruited by 
secondary rays, but on the other hand gradually get 
neutralized by combining with negative corpuscles. For if 
this were the case the rays which strike the screen near O are 
not, as in the previous case, rays which have been produced 
near the ends of the electric and magnetic fields, but are rays 
which have been neutralized soon after entering these fields. 
As the deflexions of such rays are due to the forces which act 
on the charged particle immediately after it leaves the tube 
and enters the space between the plates, the effect of pushing 
the magnetic field forward away from the tube will be to 
diminish the magnetic force on these particles while the 
electric force is unaltered: this will clearly tend to make the 
luminous band more nearly horizontal than it was before 
the magnets were pushed forward, and we see from fig. 5 that 
this is just the effect produced. 

The upper band also differs from the lower one in not 
having, so far as I have been able to observe, any negative 
portion connected with it. 

The difference between the properties of the rays which 
constitute the two bands is also shown by the following 
experiment. ‘'wo systems of magnets and parallel plates 
instead of one are placed between the cathode and the screen, 
the fields in these could be excited separately. The deflexion 
due to the magnet next the cathode is horizontal, that due to 
the magnet next the screen vertical. The electric fields are 
at right angles to the corresponding magnetic fields. Suppose 
that the magnetic field nearest the cathode is excited, the 
phosphorescent patch will be drawn out into a horizontal 


758 Sir J. J. Thomson on 


line, the most deflected portion of which, B, will be due to 
particles which were charged when they passed through the 
cathode. Now let the magnet next the screen be excited, 
the appearance on the screen is as in fig. 6; those rays 


Ke. 6. 


which were charged when they passed through the first field 
und were deflected by it are still further deflected along the 
line BC, but in addition to this the stream of neutral 
particles as it passed between the two magnets has pro- 
duced new secondary rays and these are deflected along OA. 
Thus all the rays which were charged when they passed 
through the cathode are found on the line BC, while OA 
consists exclusively of those which have been produced or 
which have acquired their charge after they left the first 
magnet. If now we put on the electric field in the system 
next the screen, we find that, at low pressures, the portion 
BC, which consists of rays charged when they parsed through 
the cathode, is broken up into “the two bands of which we 
have been ‘speaking, and which were seen when only one 
system of electric and magnetic forces was used. On the 
other hand, the band OA, due to rays which were produced 
nearer the screen than the first magnet, does not biturcate 
but consists of only one branch for which the maximum 
value of e/m = 10*. The appearance of the phosphorescence 
is shown in fig. 7. 
Fig, 7. 


OBR 
. C 


T have hitherto rita only of two bands, but when the 
pressure is low there seem with these large tubes to be para- 
bolic bands corresponding to every gas in the tube. By using 
very sensitive screens I have been able to detect the bands 
corresponding to hydrogen, helium, carbon, air, oxygen, 
neon, and mercury vapour. “The appearance on the screen 
when there are several gases in the tube is almost like a 
spectrum, and I think this effect may furnish a valuable 
means of analysing the gases in the tube and determining 
their atomic weights. There is a band on the screen corre- 
sponding to a value of e/m about +x 104, due to the air in 


Rays of Positive [Hlectricity. 759 


the tube ; the'arrangement I was using was not suitable for 
applying the most intense magnetic fields and I could not 
detect that this spot was double, with one constituent 
corresponding to the atom of nitrogen, the other to the 
atom of oxygen. When CO was put into the tube, however, 
the band in this region was clearly double although the 
constituents were very close together, one constituent I 
suppose corresponding to oxygen the other to carbon. 

One interesting feature in these experiments is that the 
bright spots on the bands are all in the same vertical line, 
showing that the electrostatic deflexion is the same for them 
all, and therefore that this energy of the particles which 
form the bright spots corresponding to the different gases 
is due to a fall through the same potential difference. The 
velocity of the rays forming these bright spots varies with 
the potential difference between the electrodes. 

The bright spots come I think from the negative glow at 
the outer boundary of the dark space; they are weakened 
by any arrangement which prevents the portion of the 
negative glow straight in front of the cathode having free 
access to the cathode. Thus, if the anode A is a disk placed 
in front of the cathode, the spots do not appear unless the 
anode is pushed back so as to be outside the dark space; 
the continuous band due to the secondary radiation is, 
however, well developed when the anode is put forward. 

Another interesting feature of these bright bands is that 

; some of them have negative tails connected with 

Fig. 8. them while others have not. This is shown in fig. 8, 

which represents the appearance in a tube con- 

" taining mercury vapour, air, helium, and hydrogen; 

i a, b, c, d, are the spots corresponding to these 
substances, the spot f is on the part due to secondary 
radiation: it will be noticed that this secondary 
radiation has a negative tail, there are no tails 
corresponding to the lighter elements, but the air 
and mercury bands have a well developed tail. 

The details of the measurements of the values 
of e/m for the different elements are given at the 
end of this paper; it may be noted here, however, 
that with the exception of hydrogen all the charged 
particles of the different gases seemed to be atoms 
| and not molecules of the gas. In working with 
es. the heavier atoms it is desirable to have very 

“ey intense magnetic fields, otherwise the magnetic 

deflexion is very small. I am making arrange- 

ments for experiments in which the magnetic forces will be 
much greater than those I have hitherto used. 


760 Sir J. J. Thomson on 


The preceding considerations show | think that we may 
divide the positive rays into the following classes :— 

1. The undeflected rays, 2. e. rays whieh are not affected 
by electric or magnetic forces; we cannot determine directly 
the velocity or the value of e/m for these rays. 

2. Secondary rays produced by the rays (1). As the rays 
of the first type pass through a gas and collide against the 
molecules they produce secondary rays; whether they do 
this by splitting up themselves or by dissociating the 
molecules against which they strike, is uncertain. The rays 
of this class have a constant velocity 2x10° cm/sec. 
roughly; whatever may be the potential difference between 
the electrodes, they have a constant maximum value of 
ejm = 10*. Atthe higher pressures and when the diseharge- 
tube is small, these rays predominate and swamp the others ; 
they get fainter and fainter as the pressure is reduced below 
a certain amount. We shall eall the rays of this type 
secondary positive rays. 

3. Rays characteristic of the gases in the tube. These are 
seen at low pressures, they produce bright spots on the 
screen ; with each spot a thin parabolic band of luminosity 
is connected, the separate bands forming a kind of spectrum 
characteristic of the gases in the tube. The velocity of 
these rays depends on the potential difference between the 
electrodes, and the value of e/m is inversely proportional to 
the atomic weight of the gas from which they are derived. 
Their kinetic energy is that due to the potential difference 
between the negative glow and the cathode, in a mixture of 
gases the electrostatic deflexion of the rays from each gus is 
the same. 

The retrograde rays which start from the cathode and 
travel away from it in the same direction as the cathode rays 
belong to classes (1) and (2). I have never seen the bright 
spots characteristic of class 3 in the retrograde rays. 

In addition to the positively charged rays there are 
negatively charged ones of type 2 and in some cases of 
type 3. The different gases show great variations in the 
brightness of the negative tails connected with the rays 
peculiar to the atoms of the element, some elements show 
the negative tail readily while I have never seen it with 
others. 

If we suppose that the undeflected rays are formed by the 
recombination of positive and negative particles and that 
these by collision with the molecules of the gas through 
which they pass form rays of type (2), either by splitting up 
themselves or by dissociating the molecules against which 


; 

a 
J 
¢ 


Rays of Positive Electricity. 761 


they strike, we can explain why the velocity of these rays 
should be independent of the potential difference between 
the electrodes inthe tube. For in the first place, the positive 
and negative charges will not unite unless their relative 
velocity falls below a certain value which does not depend 
upon the strength of the electric field, and in the next place 
if the velocity were less than a limiting value they would 
not dissociate themselves nor could they dissuciate other 
molecules by collisions when moving through a gas. The 
first condition gives a superior limit to the velocity, the 
second an inferior one; and both are independent of the 
strength of the electric field. 


I shall now proceed to give the details of the measurements 
of the values of e/mandv. These constants were determined 
by measuring the magnetic and electrostatic deflexion of the 
rays. If y is the deflexion due to the magnetic force, ¢ the 
charge on the particle and v its velocity, 


i 
e 
pala) Ron \ ae 
y =< (| (ide 
where « is the distance, measured along the undeflected ray, 
from the end of the tube through which the rays enter the 
magnetic field, H the magnetic force at the point «, and J 
the distance of the screen from the end of this tube. The 


1 
value of {, @-#)Hae was determined by measuring the 
0 


magnetic induction through a triangular coil with its base at 
the end of the tube and its apex at the screen (see Phil. 
Mag Nov. 1909). Jf is the number of turns in this coil, 
d the base and / the perpendicular from the apex on the 
base, I the magnetic induction through the coil, then 


J 
ae ei) Get y istry 
iene 


hence if we know I we can deduce the value of the integral ; 
the coil was made so narrow that for a given value of w the 
magnetic force was constant over the coil. 

The induction was measured by means of a Grassot 
fluxmeter, using for the sake of greater accuracy the de- 
flexions of a beam of light reflected from the back of the 
instrument instead of the usual index and scale. 

The fluxmeter was standardized, (1) by measuring by 
means of it the induction through one of a pair of coaxial 
solenoids when a known current was broken in the other, 


762 Sir J. J. Thomson on 


the coeflicient of mutual induction for these coils had been 
carefully determined by Mr. Searle; (2) by means of a 
Duddell induction-meter which had been standardized at the 
National Physical Laboratory and which was kindly lent to 
me by the Cambridge Scientific Instrument Co. The two 
methods gave results : agreeing within less than 1 per cent, 

With regard to the electrostatic deflexion we have to allow 
for the irregularity of the field near the edges of the plate ; 
the case is one for which a complete solution is given by the 
Schwartzian transformation 


t , 
= rare | where <= a+1y, 


dt 
or x+iy = C(t— log (1+t)+ 77), 
dw B 
ieee where w= +7, 
or bt+iv = B{log (1+t)—ir}, 
where 7 = Ca is the equation to one plane and y =—Cr 


to the other, y =0 is the plane midway between them ; 
vw is the potential and @ the current function, 2Ba the 
difference of potential between the plates. The range of ¢ 
over one of the semi-infinite planes and the plane midway 
between them is shown in fig. 9. ¢ ranges from +2 to 0 
on the upper, from ¢ == 0 to --1 on the lower surface of the 


Tig. 9. 
£=0 L=+00 
a a 
=O C= -/ 
Z=00 Cz-s 
semi-infinite plate, and fron t=—1 to. t=—« on the 


plane midway between the two plates. 
We shall suppose that the undeflected path of the partials 
is in this median plane. The equation of motion is 


ad 
m Hit a ee 
aaeel gd athe 
or approximately mv’ a= Ye, 


where Y is the electric force perpendicuiar to the plates, 


_ dp _ dd 
ie dy ~'dz* 


Rays of Positive Electricity. 763 


Hence 


hence 


d ’ 
mv? =e(dp s dy) 


where oe is the value of — dy J ata point P 5d; 1s the value of 


da da 
dat P and dy the value of ¢ at the place where the rays 
leave the narrow tube inserted in the cathode ; the value of 
4 at this place is assumed to be zero. Hence if y is the 
e 


displacement of the particle on the screen, 


=r. 
mv-y = ( e(dp— Golda, 
0 


—l 
= e( bpd 2 elo, 
0 


where 1 is the distance of the sereen from the end of the 
tube. Aiong the median plane 


dt Pee 
hp= Blog (141%); 


hence 


ay } 
ef bpda= BOY log (1+ ¢)dé 
: | - 
=cBC| (1+2) log (1+ 4) — (141) log? (1+) |°, 
where A refers to the sereen and 0 to the end of the tube. 
Hence 


mv? 


—— y= BC (1+¢) log (1 +t) — (1 +4) — $log? (1+8) |° + Blog (1+ ty) 


If the distance of the end of the tube from the edge of 
the plates is a considerable multiple of the distance between 
the plates an approximate value of to is —1. 


764 Sir J. J. Thomson on 


Let f)= —1~—€, let b be distance from the end of the tube 
to the edge of the plate, then 


b = C(—1—£&— log &), 
an approximate solution is 
E hs (Ae) 
log (1+t)= -(" ), 


If dis the distance of the screen from the edge of the 
plate, ¢, is given by the equation 


—d=C(t,— log (1+t,)); 


and when d is large compared with C, we can easily get a 
solution of this equation by successive approximation. 

Two sets of plates were used in the course of the experi- 
ments. For one set 2°5 em. long and *2 cm. apart, 


oo aa he bien roe 
ve 2°D, 
fa 2 


for these we find ‘ 
t, = —189°-46 log (14+ to) = —79°5, 
this gives 


y= <x 4, 


where X is the potential difference divided by the distance 
between the plates. 

For the second pair of plates, which were 5:0 cm. iong 
and 3 cm. apart, 


Opec ka, 
oe o0, 
= Die 
is 
for them 
t, = — 73°42 log (1+t)=—105°7, 
hence 


y=—,X 32-2 


We shall now proceed to consider the values of e/m for 
the different types of rays. First, with regard to the 
secondary rays. The values of e/m were measured when 


; 
¢ 
, 


Rays of Positive Electricity. 765 


there was a well-marked spot which was visible on both the 
positive and negative side (this is the spot f in fig. 8). 
When the conditions were most favourable to accurate 
measurements, it was found that increasing the potential 
difference between the plates from 100 to 200 volts increased 
the horizontal deflexion of the spot by 3 millimetres when 
the second system of plates was used. While an increase of 
3 millimetres in the vertical deflexion was produced by 
increasing the current through the electromagnet by 
1 ampere. The measurements of the magnetic induction 
by the fluxmeter showed that this increase in the current 
corresponded to an increase of 5:05 x 10° in the value of 


= 


L 
\ (J—x) Hdz, 
0 


a 


hence we have 


e ae 
3=— x5:0d x 16°, 
mv 
e Gre 
*3= — x—> x 32°72 
mv ‘3 ; 
as as Br 9. 8 
giving 2 KO 


e/m= 1:24 104. 


It was found that the values of e/m for this spot always 
came out a little greater than 10*, and as the spot was not 
quite at the extreme end of the straight band of phosphor- 
escence due to the secondary rays, the value of e/m for the 
rays at the tip of this band would be still greater ; for the tip 
the values of e/m ranged up to 1°5 x 10*, but as the tip is 
somewhat ill-defined the values of e/m for it could not be 
measured with the same accuracy as when there was a spot. 
The larger values of e/m were more frequent for the nevative 
secondary rays than for the positive ones; these larger values 
would be accounted for if some of the particles had acquired 
a double charge for part of their course. 

For the spot e the magnetic deflexion for a current ‘of 
2 amperes through the electromagnet 


} 
(value ot | (J—x) Hdz=1°'1 x 104) 
0 


was 4:4 millimetres, and for 200 volts an electrostatic 


766 Sir J. J. Thomson on 


deflexion of 3°5 millimetres. This elves 


‘A4-=— x1-01x 104, 


mv 
its e 2x ye 
°35=—7 x —— x 322; 
mv" 3D 
or v= 2°66 x 10°, 


e/m= 1°16 x 104. 


The value of v for this spot depends upon the pressure in 
the tube. The spot d had the same electrostatic deflexion as 
e, so that the values of e/m for the spots d and e will be as 
the squares of the magnetic deflexions. 

The corresponding magnetic deflexions for d and e and 
the square of their ratio is given in the following table : 


Deflexion of e. Deflexion of d Square of ratio. 
oD 2°9 Poe 
ad and 2°06 
G'S 4:7 2°09 
6°0 4 Zao 
7°U o 1°96 


Thus the value of e/m for d is half that for e; hence if the 
charges are the same, the mass of the carriers producing the 
spot d is twice that of those producing e, hence we ascribe d 
to the hydrogen molecule. 

The spot ¢ is the helium spot, and the value of e/mas I 
showed in my earlier paper is } that of the spot e. 

We can compare the mass of the carriers for the spot } 
with those of d by comparing the magnetic deflexions, the 
following are corresponding values: 


Spot d. Spot 0. Square of ratio. 
‘0 33 v4 
4°7 1°8 6°8 
9°2 35 69 
g°9 2°4 78 
8°0 3 (pa 


Thus if the charges are the same, the mass of the carriers 
of bis about seven times that of d; if, as we supposed, the 
carrier of d is the hydrogen molecule, then the carrier of } 
will be an atom either of nitrogen or oxygen. I am in- 
clined to think that this is a double spot and will be resolved 
by the application of stronger magnetic fields. 

When the air in the tube was replaced by CO there was a 
spot in approximately the same position as 6, on increasing 


ae - 


S patel an 


Rays of Positive Llectricity. 767 


the field it was resolved into two with magnetic deflexions 
6°0 and 5:3 millimetres ; the square of the ratio of these 
deflexions is 1°28, the ratio of the atomie weights of O and C 
is 1:33, which agrees with the preceding value within the 
accuracy of the experiments. ‘The spot a was compared with 
b, the corresponding magnetic deflexions are : 


b, de Square of the ratio. 
20 1°5 i 
20 Lo 13°6 

Mean. 4... 12°35 


For mercury vapour the square of the ratio would have 
been 14 if the spot b were due to nitrogen, 12 if it were due 
to oxygen. The deflexion of the spot a with the magnetic 
force available was too small to admit of accurate measure- 
ment, but there can, I think, be little doubt that the spot a 
is due to mercury vapour. Jt disappears very quickly when 
liquid air is put around some charcoal in a side tube. 

Thus we see that on the assumption that the charges are 
equal, we see that all the carriers with the exception of those 
for spot d are in the atomic condition ; a very remarkable 
result, and one which has an important bearing on the 
dissociation of gases in the discharge-tube. It will be inter- 
esting to liberate the different elements from compounds of 
different types when they have different valencies, and from | 
earbon compounds where the bands are different, and see 
whether the value of e/m remains unaltered. 

The absence of the negative part of the phosphorescence 
indicates a reluctance on the part of the atoms of some gases 
to acquire a negative charge ; this is also brought out by 
Franck’s discovery that in some gases from which oxygen 
has been carefully excluded the velocity of the negative ion 
was very manv times greater than when oxygen was admitted, 
while the positive ion was not affected. This indicates that 

:negative corpuscle does not readily attach itself to the 
molecules of these gases. 

I had occasion in the course of the work to investigate 
the secondary Canalstrahlen produced when primary Canal- 
strahlen strike against a metal plate. I found that the 
secondary rays which were emitted in all directions were for 
the most part uncharged, but that a small fraction carried 
a positive charge. 

I have much pleasure in thanking Mr. F. W. Aston, of 
Trinity College, Cambridge, and Mr. E. Everett, for the 
kiud assistance they have given me with these experiments. 


, | LXAXXIV. Vacuum Spectrometer. 
By Professor Augustus TROWBRIDGE * 


[Plate XVI.] 


| eae the purpose of spectroscopic investigation in the 

extreme ultra-violet region of the spectrum it has been 
found necessary, on account of atmospheric absorption, to 
employ some form of vacuum spectrometer, either of the 
mirror type or of that where collimation is effected by lenses 
of some transparent material. 

The instruments employed in the well-known work of 
Schumann and of Iyman could hardly be improved on were 
one to design an instrument for research in the ultra-violet 
region exclusiv ely, and I therefore only venture to describe 
an instrument which I have recently had constructed because 
of its wider range of usefulness. 

In research work in the infra-red region of the spectrum, 
it is customary to use a mirror spectrometer of the fixed arm 
type with a rock-salt prism and Wadsworth mirror. The 
energy measuring instruments most commonly employed use 
the bolometer or the thermopile. It is generally necessary 
to shield either of these very carefully by means of screens 
from irregular changes of temperature, and it is not un- 
common in the use of the bolometer to mount it in an air- 
tight case with a transparent window. Some observers have 
worked with the bolometer zn vacuo in order better to secure 
constant temperature conditions. 

If rock-salt be employed as the prism substance it is 
necessary to protect it against moisture, and this requires 
that the prism be enclosed in a case with the necessary 
openings for the passage of the light and suitable arrange- 
ments for preserving a moisture-free atmosphere within it. 

From the above it is evident that a vacuum spectrometer 
would be advantageous, though not of course absolutely 
essential, in securing good working conditions in infra-red 
investigations as well as in work in the ultra-violet. 


The instrument described in the present paper is designed 
to be used with either a prism or a grating, and attachments 
are provided which allow the use of either bolometer or 
thermopile for work in the infra-red or a photographie plate 
carrier for work in the ultra-violet. 

Referring to the first of the figures, which are photographic 
reproductions of the shop drawings, and are one-quarter of 


* Communicated by the Author. 


en ee 


A Vacuum Spectrometer. 769 


the natural size. Cis the main conical bearing of the spec- 
trometer—it carries on its upper end (not shown in Pl. XVI. 
fig. 1) the grating or prism-holder and near its lower end a 
large divided circle D. Readings are taken on this circle 
by means of micrometer microscopes MM, one division on 
the head of which is one second of are. Less accurate read- 
ings may be rapidly obtained from the setting of the drum 
N mounted on the axis of the worm W, which engages the 
gear-wheel WW rigidly attached to the cone and divided 
circle. The worm W is mounted on the bed-plate B, which 
is bolted to the three legs of the instrument. The worm may 
be thrown out of gear by a suitable mechanism. 

Pl. XVI. fig. 3 is an elevation of the upper part of the 
instrument, showing a section of the evacuated region 
and arrangement for leveling the grating or prism-holder I. 
This holder may be centered by means of the screws c, and 
it may be rotated about the axis of the cone C’, which is 
approximately coaxial with the cone CO. 

Pl. XVI. fig. 2 shows a horizontal section of the instrument 
through the axis of collimation ; SC is the carrier of the slit 
—this brass casting which carries the slit SS’ is provided with 
a window W of suitabie transparent material and the necessary 
conical plugs to admit of adjustment of the slit from the 
outside as regards height, width, and position. The details 
of this are not given as I have followed the construction 
devised by Schumann. The slit carrier SC is held on the 
collimator tube O by means of the ground conical gearing 
provided with the screw-ring R;. ‘The collimator tube is 
provided with appropriate diaphragms, and is soldered with 
the main bronze casting excentrically as shown in the figure. 
M, and M, are concave mirrors, of silvered glass for the work 
in the infra-red, and of speculum metal for ultra-violet work, 
so placed as to render the beam falling on the grating 
parallel, and then to bring the image of the slit on the 
bolometer strip mounted in the cone BC. 

In case a prism be used instead of a grating the mirror M, 
is placed at M,', and the bolometer case BC and the cap B’C’ 
exchange positions. The details of construction of the carriers 
of the mirrors M, and M, are given in the margin of fig. 3. 
A diagram of the electrical connexions of the bolometer is 
givenin the margin of fig. 2. ,...b, represent the bolometer 
strips 1-4, and 4 and 3 the balancing coils of manganin wire 
which are bifilar wound and occupy the capsule c...c shown in 
the bolometer case BC. For final adjustment of balance one 
of the coils is shunted with the high resistance R’. 

A glass window is provided at the back of the bolometer 


Phil, Mag. 8. 6. .Vol. 20:.No. 118. Oct 1910. aH 


77 Prof. R. W. Wood on the 


case so that the exposed bolometer strip may be viewed at 
any time by means of a low power microscope (shown in 
fig. 4). 

For work in the ultra-violet spectrum a plate-carrier case 
replaces the bolometer case BC. ‘The details of this plate 
carrier are not given, as I have here also followed the 
construction devised by Schumann. 

The instrument has been in use for several months as a 
spectrobolometer, and has proved in every way satisfactory. 
The gain in speed of observation over that possible with a 
bolometer subjected to disturbances due to air currents more 
than compensating for the loss of time in pumping out the 
instrument. 

No annoyance whatever has been experienced from “leak,” 
though of course the instrument is not perfectly air-tight. 
The ‘leak, chiefly at the rock-salt window on the slit-case, 
does not exceed one millimetre of mercury in forty-eight 
hours. In practice a Gaede pump is run continuously 
during observation at a speed just sufficient to take up the 


leak. 


LXXXV. The Echelette Grating for the Infra-Red. By 
R. W. Woop, Professor of Huperimental Physics, Johns 
Hopkins University * 

(Plate XVII 
NE of the most impurtant problems in Optics is the 
question of the distribution of intensity among the 
spectra of different orders produced by a dittraction grating. 

Practically no rigorous experimental investigation has been 

made, owing to ‘the impossibility of determining the actual 

form of the groove ruled by a diamond point on a glass or 

metal surface. It is very difficult to learn anything “from a 

microscopical examination, and it is by no means certain that 

the form of the groove wil! conform to what we believe to be 

the shape of the ruling point. It occurred to me that a 

promising method of attack would be to manufacture gratings 

with grooves of such large size as to make the determination 
of their exact form, width, &. a matter of certainty, and 
then investigate the energy distribution by means of the 
long heat-waves discovered by Rubens and his collaborators. 

By ‘employi ing the residual rays from quartz and a grating 

with 1000 lines to the inch, we should have about the same 

ratio of wave-length to grating space as obtains in the case 
of a = mae grating with 14,000 lines to the inch, and. ae 


* Communicated by the Author. 


Echelette Grating for the Infra-Red. Tin 


hivht. Gratings with constants varying from 0:1 mm. to 
‘01 mm. could be studied by means of residual rays, or 
narrow revions of the infra-red spectrum, isolated by a salt- 
prism spectrometer, and the relation between the intensity 
distribution and the form of the groove determined. Methods 
were worked out by which a groove of any desired form 
could be ruled, with optically tlat sides (a very important 
point), the angular slope of each side of the groove measured, 
and the exact nature of the ruling determined, BiGanin henter 
the metal had been forced up between the grooves, or 
whether the angle between the opposed faces was equal to 
the angle between the edges of the ruling knife. This by 
no means follows, as the ruling of groove No. 2 may force 
the metal to one side and increase the angle of slope of the 
adjacent side of groove No. 1. Gratings were finally obtained, 
which have proved so etticient in the investigation of infra- 
red spectra that it seems worth while to designate them by a 
name of theirown. They throw a large percentage of the 
energy into one or two spectra to the left of the central 
image, and show little or no trace of any energy to the right 
of it. With visible light they send the greater part into a 
group of spectra, say, trom the 12th to the 16th, or from the 
24th to the 30th order. Thev may thus be regarded as re- 
flecting echelons, of comparatively small retardation, and I 
propose the name “ echelette,” to distinguish them from the 
ordinary grating and the Michelson echelon. 

Various methods were tried for their production. The 
first were made by punching the grooves with a steel die, 
two adjacent surfaces of which had been ground flat and 
highly polished. The die was a block of hard steel measuring 
3x2x1:5 cms., and the gratings were punched with an 
ordinary milling machine, the die being clamped at the 
proper angle in a fixed position, and a polished plate of some 
soft metal pushed up against it from below. This method is 
analogous to the one used by Mr. Thorpe, in making his 
gratings for the demonstration of predominant spectra, but 
it did not give very satisfactory results. 

After considerable experimenting with various metals and 
ruling points, I came to the conclusion that sott alloys must 
be avoided, for it appeared to be impossible to cut a groove 
with optically flat sides. The crystalline structure of the 
metal caused the point to rule a groove, the sides of which 
undulated more or less, causing more or less reflexion in 
directions parallel to the grooves. 

The method tinally adopted was the following :—A sheet 
of a copper plate, such as is used by photo- -engravers. 


dH 2 


Tz Prot. R. W. Wooil on the 


for the half-tone process, was gold-plated and_ polished. 
The plates were found sutticiently “flat for the purpose and 
had a much better optical surface than anything that I was 
enabled to produce on a copper plate by grinding and polish- 
ing, for the final polishing always produced irregular undu- 
lations (possibly owing to variations in the har dness), and I 
was unable to get any suggestions from professional opticians 
accustomed to the polishing of glass and speculum metal. 
If any one has worked out a method of getting a flat optical 
surface on such metals as copper or gold, I shall be very glad 
to hear of it, as it will doubtless improve the quality of “the 
gratings. 

A carbor undum crystal was used for the ruling point, and 
the ruling, in the case of the gratings of very large constant, 
was done on a small laboratory dividing-engine by hand. 
This machine had bad periodic errors, and the best gratings 
were made with Rowland’s first machine, with a 7-tooth and 
a 15-tooth cam, which gave 2062 and 962 lines to the inch 
respectively, 

The hexagonal carborundum crystals were selected by 
breaking up a mass of the substance as it comes from the 
furnace. Specimens of these iridescent crystalline masses are 
to be found in most chemical or mineralogical museums. 
The crystals have the form sbown in figure 1, and are 


Grating Plate 


mounted as shown in the figure. The natural edges are so 
straight that they rule a groove with optically perfect sides. 
Everything depends upon the nature of the cdge and the 
angle at which it is set with respect to the direction of the line, 
2. e. the tilt forwards or back. Some edges will not rule 
properly at any angle, “chatteriny ” over the surface and 
tearing off a thread of metal. Vo metal is removed when the 
ruling is going on properly, the groove being formed by 
compression of the metal. Jf the ‘edge is properly chosen, 
mounted at the proper angle and correctly weighted, a 
beautiful groove is made with a very little élevation of the 


Echelette Grating for the Infra-Red. Cs 


metal above the original surface at the edges. The first 
gratings were ruled on copper and subsequently gold-plated, 
to prevent tarnish, but it was found that even the lightest 
polishing on the bufting wheel destroyed the sharpness of the 
edges and caused the development of strong central images. 

It was found, however, that even with an exceedingly thin 
deposit of gold (about the lightest plating ever done com- 
mercially) it was possible to rule very deep grooves without 
uncovering the copper. This solved the difficulty, and 
excellent gratings could be produced at a very smail cost. 
The copper plate was varnished with asphalt on the back to 
save gold, and gilded in as large pieces as the gilding 
establishment could handle. ‘These large sheets were then 
cut up to the required size with a circular saw. They per- 
formed fairly well optically, giving almost as good images as 
an ordinary plate-glass mirror, in spite of the rather rough 
treatment to which they had been subjected. It is important 
to instruct the gilder to do as little butting as possible. My 
first plates were spoiled by having too thick a deposit of gold 
and too vigorous buffing or burnishing. ‘The best treatment 
is the one which they give to the thinnest coats, which would 
be completely removed if polished by the methods employed 
for thicker deposits. Ifa thick deposit is given and polished 
in the usual way, the optical surface is ruined by the formation 
of undulations, though it is hard to convince the gilder that 
it is unsatisfactory. I mention these details for the benefit 
of others undertaking the manufacture of these gratings, 
for it took me nearly a week to convince the gilder that 
he could be taught anything about the nature of metals 
and how they should be treated. To obtain a better optical 
surface, or rather a flatter one, I had a polished flat plate of 
speculum metal, such as is used for making Rowland gratings, 
silver-plated and pclished. The circumstance that the first 
plate which I placed in the hands of the plater flew into three 
pieces as soon as he put it in the hot alkaline solution which 
they use for cleaning thin metalwork, convinced him that there 
was something about metals to be learned, and he was more 
willing to take advice thereafter. By this method it was 
possible to get a beautiful optical surface of soft metal, in 
which the grooves could be out. So far as I could see by a 
rather superficial examination, the optical perfection of the 
surface had not been materially attected. 

The angle of the ruling edges of the carborundum hexa- 
gonal plates is 120°, consequently the sides of the groove 
make approximately this angle. By placing the crystal in 
various positions we obtain grooves of various shapes, one 


CTA Prof. R. W. Weod on the 


side, for example, sloping at an angle of 12° with the original 
surface, the other at 48°. These angles are subsequently 
determined with a small spectrometer or by simply mounting 
the gratings on a graduated circle, and observing the reflexion 
of a lamp-flame in them. In the best ones no trace of the 
central image can be seen, which is what we should expect 
if the edges of the grooves were sharp and none of the 
original plane surface remained. The sum of the two angles 
of slope did not always add up to 60°, as they should do 
if the ruling had been done with a 120° point. This is 
probably due to the circumstance that the edges of the 
carborundum crystals are usually bevelled as shown in fig, 1. 
I have not made a study of the angles at which these small 
planes meet, but it seems likely that with certain crystals we 
may have a ruling point the edges of which meet at an angle 
larger than 120°. 

Of the eight gratings which I have measured thus far, 
the angles of the edges and their sum are shown in the 
following Table :— 


thy eve 22.0029, 11.) die 
49  2io 30 18 44. 46, oie 


60 48 a2: 4d bd... Gap 54) ee 


In addition to knowing the angle of slope of the two sides 
of the groove it is necessary to determine whether they 
make a sharp angle, i.e. whether they meet in a knife-edge 
at the top, or whether there is some of the original surface 

remaining between them, or a ridge of more or less roughed 
surface due to the squeezing up of metal by the compression 
resulting from the action ot “the carborundum crystal. 

Some difficulty was found in interpreting the appearance - 
of the surface under the microscope until the following 
method, which gave beautiful results, was tried. Two electric 
Jamps were placed just above the stage of the microscope, to 
the right and left of the tube, in such positions that the 
edges “of the grooves reflected light vertically into the 
objective. A red glass was placed in front of one lamp, and 
agreen glass in front of the other. The edges of the grooves 
appeared brilliantly illuminated in complementary colours, 
with no dark region, if they met at the top, but if not, each 
pair of red-green strips was separated from the neighbouring 
pair by a dark line, due to the fact that the level surface 
between the grooves was not at the proper angle to reflect 
light from either lamp into the objective. The appearance 
of the gratings illuminated in this way was similar to that 


Lchelette Grating for the Infra-Red. 778 
of one of the screens used in the Joly process of colour- 
photography. If the grating showed no central image 
with light, it was safe to assume that the metal along the 
dark line had been forced up and the original surface 
destroyed. This was usually the case when the grooves 
were very nearly in contact. If strong central images were 
exhibited, it indicated that a portion of the original flat 
surface remained between the grooves. The width of this 
portion in comparison with that of the roughened portion 
could be determined by placing the grating at an angle 
under the microscope and reflecting light from the linear 
strips between the grooves. In this way a very perfect 
knowledge of the exact nature of the ruled surface could be 
obtained. Another method of studying the surface is to 
make a cast of it in celluloid or a paraffin composition and 
section this with a microtome. The optical method gave 
the best results, however. 

In the majority of cases the crystal was mounted so as to 
rule a groove one edge of which made an angle of 20° or 
less with the original surface. With normal incidence this 
gives us a concentration of energy at an angle of 40°, with 
practically no energy thrown off from the other edges of 
the groove, owing to the steepness of the angle. This case 
is shown in fiy. 16. The best gratings show no reflexion in 
the normal direction, t.e. they 2 give no central image. They 
vive, however, a very good reflected image of one’s face, 
when held at an angle of 20°, the i image being uncoloured, 
but slightly diffused by diffraction in a direction perpen- 
dicular to the grooves. The image is so sharp, however, 
that the pupil of the eye can be seen without difficulty. 

The gratings behave, with infra-red radiation of wave- 
lengths, above, say 3m, precisely as an ideally perfect 
grating, that is they give spectra similar to what we should 
have with an ordinary grating which threw practically all 
of the light into one or two orders on one side of the 
central image. 

With visible light their behaviour is most curious and 
interesting. The centralimage is usually absent, and we geta 
blaze of light when the grating is turned at the proper angle. 
Witha symmetrical groove the blaze is seen on both sides, 
at angles of 45° for normal incidence. ‘This blaze we may 
term the oblique image. 

It the source of light is white, a lamp flame, for example, 
the appearance is as shown in fig. 2d (Pl. XVIL.), which is for 
a grating with a constant of ‘05 mm. The position of the 
central image is indicated by an arrow. It is very faint or 


776 Prof. R. W. Wood on the 


barely visible, however. To the right and left are the 
oblique images, but very slightly broadened by diffraction 
owing to the width of the reflecting edges, which in this case 
make equal angles with the surface, 

In fig. 2a we have the appearance of things with a 
grating “of constant ‘0123 mm. The central image is in- 
dicated by an arrow, and is bordered on each side by the 
ordinary grating spectra, which are close together on account 
of the coarseness of the ruling. They are much fainter than 
I have indicated on the plate. W ell to one side, at an angle 
of about 40° with the normal, we see a very bright and 
greatly broadened white image of the flame, accompanied by 
Jateral spectra. These are shown by a coloured plate in the 
forthcoming edition of my ‘ Physical Opties.’ 

These are not grating spectra, but the first class spectra 
(as Fraunhofer termed them) due to a single slit, or in this 

case to a single reflecting edge of a groove. With a sodium 
flame the appearance is as shown in fig. 2b. We have in 
this case three orders of spectra in the region occupied by 
the central maximum of the spectra of the first class. Their 
order is indicated below. One or two orders to the right 
and left of this group are absent, since they fall in the 
region of the minimum due toa single slit. They are the 
“ absent spectra” of grating theory. Other groups of orders 
appear in the regions occupied by the first class coloured 
spectra, their intensity being much less, however, than the 
intensity of the ones falling within the region of the central 
maximum. ‘The existence of these images of the soda flame 
shows us that the perfection of ruling is such that rn 
ference, with a path difference of about 30 wave-lengths, 
still taking place. In other words, our grating is me as 
a reflecting echelon with steps 15 wave-lengths in height. 
This was observed only in the case of the gratings ruled on 
the Rowland machine. Those ruled on the small laboratory 
machine by hand showed only a confused jumble of over- 
lapping images, which formed an almost continuous band of 
yellow light. ‘he grating constant was ‘0123 mm. in the 
case represented in figs. 1 and 2. With a larger constant 
the central maximum of the spectra of the first class was 
narrower and brighter, the lateral rainbow coloured fringes 
being less in evidence. 

The width of the region in which we have these maxima 
and minima of the first class becomes less as we increase the 
width of the reflecting steps of the grating. In the case of a 
grating of such a small constant as °0123 mm., this region 
of diffraetion of the energy from each individual element 


Echelette Grating for the Infra-Red. 777 


covers a range of fully ten degrees, embracing as many as 
12 or more orders of second class spectra. Tn the case of 
the Michelson echelon, the width of the step is from 0°5 to 
1 mm., and the range of diffraction is so small that but one 
or two orders of spectra are included within it. By the 
study of these echelette gratings we can pass eradually from 
the case of the ordinary grating to that of the echelon. 

The results appear to me to indicate that with a simple 
groove, such as we have here, we cannot secure a concen- 
tration of light in a region narrower than the diffraction 
range from a single reflecting element. This question will 
be more fully discussed in a subsequent paper treating of 
the energy distribution among spectra of different orders 
produced by these gratings, with visible light and very long 
heat waves. 

Some of the gratings, with a constant of ‘0123 mm. gave 
strongly coloured i images, and lateral spectra of low or der in 
which a certain colour or colours were wholly absent. The 
first order spectrum on one side, for example, may contain 
no yellow-green, a broad dark band bisecting the spectrum. 
A third order spectrum may have two dark bands, one in the 
yellow, and another in the greenish-blue. 

In one case the oblique image or the central maximum of 
the spectra of the first class, instead of being white, was 
distinctly blue, while the maxima immediately to the right 
and left of it contained only red, orange, and yellow light, 
as shown in fig. 2¢ (Pl. XVIL.). 

This curious distribution of colour was observed in the 
case of one grating only, and its explanation gave a good 
deal of trouble. To explain it we must devise some type of 
reflecting element which will give, in the case of red light, 
zero illumination at the centre of symmetry, with strong 
jateral maxima, and with blue light a strong maximum at 
the centre, bordered by minima which occupy the positions 
of the red maxima, and maxima in the positions of the red 
minima. This can apparently be brought about only by an 
element consisting of two parts, in other words a double 
reflecting strip, with a half-wave retardation for red light, as 
in the case of the laminary grating. The central maximum 
will vanish in this case for red light, as can be easily seen by 
constructing the diffracted wave-fronts. If blue light is used 
the retardation becomes very nearly a whole wave, and we 
have the centre of the system bright. An examination of 
the grating with a microscope showed that there were in 
fact two reflecting strips in contact which together formed 
one side of. the groove, the grating being bnilt up of paired 
reflecting elements separated by inoperative strips of about 


778 Mr. J. Satterly on the Absorption of 


the same width. Just how the carborundum erystal managed 

to rule such a groove I am unabie to say. <A coloured 

See of this very remarkable set of spectra will be found 
‘ Physical Opties.’ 

ih ok cases will be more fully discussed in a subsequent 
paper, in which a full report of an inv estigation made in 
collaboration with Prof. A. Trowbridge, of Princeton, with 
his remarkably perfect vacuum spectrometer will be given. 
This investigation has shown that these gratings give far 
higher resolving powers in the infra-red than have ever been 
available previously, comltined with great efficiency. The 
emission band of CO, from the fae of a Bunsen burner, 
which has been ied 9 only as a single band up to thin 
time, was easi y resolved into three or possibly four cum- 
ponent bands. 

Further experimenting will probably improve the quality 
of the gratings and open up a large field of work in the 
infra-red revion. The gratings y ield excellent replicas which 
can be mounted on flat plates ‘of glass and gold-plated by 
the cathode discharge. ‘I'he replicas will very possibly have 
flatter surfaces than the original gratings, if properly 
view asian 


— AS nes | 


LXXXVI. Some capesataal on the Dasari of Ree 
Imanation by Coconut Charcoal. By JOHN SATTERLY, 
A.R.C.Se., B.Se., BA., St. John’s College, Cambridge*. 


URING the atctibddtod of some experiments f by the 
author on the measurement of the amount of radium 
emanation in atmospheric air by the charcoal absorption 
method, the following interesting points came up for con- 
sideration :— 


(a) Is the amount of emanation absorbed from the air 
always the same fraction of the total amount in the 
air whatever that amount may ce other experimental 

conditions remaining the same ? 

() In the case when the air flowing to the charcoal 
contains a constant percentage of emanation, is the 
amount absorbed by the charcoal proportional to the 
time the air-.urrent is flowing, or does the charcoal 
show signs of saturation ? 

() Does the amount of emanation absorbed from the air 
depend on the humidity of the air ? 

(d) What is the percentage of emanation absorbed in any 
particular case ? 

* Communicated by Sir J. J. Thomson, F.R.S. 
+ See Phil. Mag. Oet. 1908 and July 1910, 


_ 2. 
— 


eee 


Aa ee 


Radium Emanation by Coconut Charcoal. 779 
ke 


Experiments to test the first point were made in Dec. 1907- 
Jan. 1908 and July-Aug. 1908. The apparatus used was 
that described in my earlier experiments *. Two radium 
solutions were made up as follows :—126 c.c. of a certain 
radium solution was taken and divided into two parts of 
43 and 83 c.c. (it was intended to be 42 and 84), so that the 
radium contents were as 1:1°93. Hach part was made up to 
136 c.c. and placed in exactly similar bottles fitted with 
inlet and outlet tubes so that air couid be bubbled through 
the solutions. Three exactly similar charcoal tubes, A, B, C 
(porcelain tubes 60 cm. Jong, 1°6 cm.’ cross-section, central 
foot filled with 39 gm. charcoal) were joined up in a circuit 
as shown below :— 


; TT GVA ees oe Tube A—Gauge A, 
From y : : ; : ae To water 
= Bottle.) >—Strong Tad ee end? Rube B Cie Bas um 
Peicide . solution. rying-tube. Be vane 


\ Weak radium. Calcium-chloride__ 
solution. drying-tube. 


Tube C——Gauge C / 


Air was drawn through the three tubes by a water-pump. 
The air-streams through the tubes were measured by means 
of three gauges and adjusted to be of the same strength. 
After the air-streams had been flowing for some hours 
they were stopped and the tubes tuken and heated. The 
amount of emanation the charcoal had absorbed was then 
measured in the usual way. 

Let H=amount of emanation in the quantity of air that 

passed along to tube A, 

and H=amount of emanation generated by the weaker 

solution in the given time of exposure. 

Then the amounts of emanation arriving at tubes C and B 
were H+ EH and H+1:93E respectively. Let the amounts 
of emanation caught by the tubes A, B, C be denoted by 
nH, n\(H+1:93 E), n.(H+H). Then, if 


ty HP Vio a) ahs) 195 
n(H+E)—nH ~ 1 ’ 


it follows that n=n,;=n.; i.e., the same fraction of the 
emanation was absorbed in each case. 

Throughout the paper the amount of emanation is expressed 
in the same arbitrary unit. The numbers are the leaks per 
minute produced in my testing vessel by the emanation and 
read on the scale provided to the electrometer. 


* See Phil. Mag. Oct. 1908 and July 1910. 


780 Mr, J. Satterly on the Absorption of 
The following are the results :— 


(1) Dee. 20, 1907. Duration of exposure 2 hrs. 
Air-stream="5 litre per min. — 


Tube A, nH = ; 
. ws H+1:93E)—nH. 26-7 
, B, n(H41-938)= 27-0 n(H +1935) —oi . 2 
sae ) q n(H--E)—nH 19a 
mnie ¢ n(H+E) = 20-0 
(2) Jan. 21, 1908. Duration of exposure 80 min. 
Air-stream = ‘5 litre per min. 
Tube A, nH = sf 
93 2) .“ 
, Bi 8H) = 136 3, MURt 13) 2h 
C n(H+E) = 9-4] Pl hry ges ty 


From (1) and (2) it follows that with the given solutions, 
n, 4, NM, are not equal. Weaker solutions were now tried. 
The strong solution was mixed with two-thirds of the weak 
and new solutions were made up, one twice as strong as the 

her. Call tl 2K, and B 
other. Call them 2K, and H,. 


(3) July 22, 1908. Duration of exposure 23 hrs. 
Air-stream ="5 litre per min. 


Tube A, nH as i H42u)—nH 149 
kl n.(H+2K.)—nH A 
Bb , 25 — Qh a ee 
a Be, : % M8) n(H+E,j)—nH 100 
99 9 2 1 sare 


Again made new solutions: one-third of E, was taken for 
the new weak solution and two-thirds of I, for the new strong 
solution. 


(4) July 31,1908. Duration of exposure 2+ hrs. 
Air-stream =°5 jitre per min. 


Tube C, nH Br is'hs ee (H42E)—nt 5 
Qa a — Va ER Nie ay gt = pi Ce as 
» By n,( H+ 3 E,) a9 eee | n (A +41 )—n 2d am 
RY ee 8 


(5) Aug. 5. Same solutions used. Duration of exposure 2 brs. 
Air-stream="d litre per min. 


Tube A, ae 7 i nia! Tilak ea 
‘ 2 ae é ~~. DoS eS eee 
” CG; n,(H+2 *) Se horde uw (H+4l,)—v0H 2-4 ris 
9 B, n( HL + x=) = 23 


‘lhe results of Experiments (4) and (5) show that for 
suiutions (f the strengths used and for the other experimental 


t 
o 
t 
Don 
i 
Uae 
Pils ; 


—— * =. 
Eee 


Radium Emanation by Coconut Charcoal. 781 


conditions, the amount of emanation absorbed by the charcoal 
is practically proportional to the emanution content of the 
air sent through the charcoal. For stronger so‘utions as 
used in (1),(2), (3), the amount absorbed does not increase 
in the same proportion as the emanation content of the air ; 
it certainly looks as if the charcoal were approaching 
saturation. 

The strength of the solution 3H, is very nearly the same as 
that of a solution containing 3°14 x 107°? em. radium which 
was given to the author by Professor Rutherford; for that 
solution gave me a reading 2°5 for a 2-hours run*. The 
solution 1H, therefore contains about 3x 107° gm. radium, 
and the emanation it would yield in 24 hours would have a 
velume of 3x 10-“ c.c. It is hard to imagine that charcoal 
would be saturated even with a volume many hundred times 
this+ ; but it must be remembered that an air-stream is 
passing through the charcoal all the time and that this air 
has two effects: (1) it is absorbed by the charcoal, thus 
leaving less room for any other gas, and (2) it tends to blow 
out any otiuer gas that has been absorbed. 

It follows that if for solutions of strengths 3x 107° and 
6x 107° gm. radium the amount absorbed is proportional to 
the strength of the solution, this proportionality would also 
hold for weaker solutions and also for the radium content in 
the air, thus justifying the method of calculation employed 
in my paper in the Phil. Mag. of October 1908. 


ae 


Experiments made to find ou! whether the fraction of the 
emanation absorbed, when the emanation is supplied 
from a constant source by a steady stream of air, ts 
andependent of the tume of exposure. 


In this series of experiments two silica tubes, A and B, 
(each 60 ems. lorig, 8°0 sq. cm. in cross-section, and con- 
taining about 139 gm. of coconut charcoal in the central 
foot) were coupled up as shown in the accompanying diagram 


i woe ed tee A or B—-Gauge A or BY % 
Outside see a T 
air. We? NX — > — 
NRadium Solution—Tube B or A—-Gauge B or ee pump. 


* Phil. Mag. Oct. 1908, p. 599. 

+ At 10°C. one gram of coconut-charcoal absorbs 3X10-5 c.c. of 
radium emanation (Rutherford, Manch. Lit, Phil. Soc. Dec. 1908). This 
is, of course, a statical result. 


782 Mr. J. Satterly on the Absorption of 


with a radium solution in series with one of the tubes, and 

exposures were made of ditferent periods but with air-streams 

of the same strength (‘5 litre per minute), The radium 
: 3°14 

solution used contained —— x107* om. of radium. Some- 


r 


times the solution was in series with Tube A, sometimes with 
r . 

Tube B. In calculating the last column, allowance has been 
made for the fact that Tube B absorbs about 10 per cent. 
more than Tube A*. 


TABLE I, 
| | Air alone. Air+ Solution. 
le.” Women Boe, it Wide 1h toa 
1909. TRAP ) g | ue aa | Emana- ! m “y i | wi) ia Bg 
| | pi aa in | ROB (thr HBS, in | tion from the 
| litres. | poner litres. | caught. solution. 
AS RES eee Erie (pasteles ee dae 
| Oct. 26-27.| 22thrs.| A | 633 | 11 || B | 660 | 47 | 47-19=85 
eS a ie a A | 640 | 41 | 41— 8=33 
, 23-29,/15 , | A | 452) 7 | Bl 448] 86 | 386— 7=29 
| Nov.2 i...) 7 » | A | 212 | 18 | B | 24 |. 39 4) ogee 
, 23../2 , | B |e | 22 | « | 67] 52 | 52-20=82 
|. g4.sja , | A’ | 687.).26, |B.) 681]. 7-2. | Foe 
gay AOE boa i vs ee 6 | A | 218) 20 | 20—jepeiao 
| 3 Pe Pa Vos 11 B | 209 |} 28 | 28—11=17 


From the figures in the second and last columns the 
following curves (fig. 1) have been plotted. If for exposures 
of all the periods named the fraction of emanation absorbed 
was the same, the curves would be practically straight. 
(There would be a slight falling away from the straight line 
owing to the decay of the emanation: this, however, is very 
slight, as is evidenced by the dotted line in fig. 1, which 
represents on an arbitrary scale the growth of radium 
emanation from a quantity of radium.) 

Sim‘lar experiments made in July and August + with a 
3-lours exposure anda solution of 3:14 x 107° gm. radium 
gave 4°5 for tube B and 4°4 for tube A. Therefore, assuming 
as is proved in I. that the amount absorbed from these solu- 
tions in a short time is proportional to the strengths of the 


* See Phil. Mag. July 1910, pp. 15, 17. 
Tt Phil. Mag. July 1910, p. 27. 


Radium Emaaation by Coconut Charcoal. 783 


solutions, the amount absorbed from the one one-fifth solution 
in 3 hours should be ‘9 for tube B and a little less for tube A. 
This is in good agreem2nt with fig. 1. 


& 12 1G 
EXPOSURE (Hours). 


The results of these experiments show that in the early 
stages not much of the emanation is allowed to pass through 
the charcoal unabsorbed. In the later stages, however, much 
of the emanation is allowed to pass. 

From fig. 1 it follows that if we suppose complete absorp- 
tion to oceur for exposures of 3 hours or less, then B for a 
21-hours exposure absorbs about 62 per cent. of the emanation 
sent through it. The state of uffuirs is probably very similar 
to that discovered by McBain * for the absorption of hydrogen 
by charcoal, viz. that the absorption is twofold: a quick 
effeci—a surface condensation—being followed by a slower 
effect—a diffusion into the interior. 


ERT 
Effect of the Humidity of the Arr. 


As no drying agent was used in the experiments described 
in II., the results may be due to the fact that as time went on 
the charcoal tube attached to the radium solution would 


* Phil. Mag. Dec. 1909, 


Outer air bubbled through the solution 
and then dried by calcium chloride.) 7°8 


Outer air bubbled through the solution, 
then saturated by passing through} 7:3 
100 cm. wet cotton-wool. 


Outer air bubbled through the solution, 
then dried by calcium chloride...... 118 


784 Mr. J. Satterly on the Absorption of 


absorb more and more water, and thus its absorbing qualities 
for other gases might gradually diminish. Experiments were 
now made to test this point. The (silica) tubes A and B 
were placed in parallel, and air-streams of the same streneth 
(‘5 litre per minute) were passed througheach. The humidity 
of the air going to the separate tubes was altered as shoyn in 
Table II. In the first three experiments the source cf emana- 
tion was the air ; in the others this source was supplemented 
OD, bubbling the air through a radium solution containing 

4 of 3:14 x 10-2 om. radium. The duration of the exposure 
was about 21 hours in each case. 


TABLE II. 


} \) 


eR | 
Tuse A. Be Tuse B. 
cS 
Outer air, humidity unaltered ......... 1:5 | Outer air bubbled through 6 in. water. 
2 | oe ; 
Outer air, dried by calcium chloride, | Outer air dried by same calcium chlo- 
then bubbled through 6 in. water. 14 ride as used with A......:.s0s0seesmanee 
Outer air, humidity unaltered ......... eae, Outer air sent through 40 cm. wet 
| cotton-wool, 
Laboratory air bubbled through the | Laboratory air bubbled through the 
Rae 16°2 solution and sent through 100 cm, 
of wet cotton-wool. 
Outer air bubbled through the solu-! Outer air bubbled through solution 
ORES, SC epee: 6'8 and then dried by calcium chloride. 


then dried by calcium chloride, 


wet cotton-wool. 


From these results it is clear that, allowing for the fact 
that under the same conditions tube B nearly always absorbs 
about 10 per cent. more emanation than tube A, the amounts 
ef emanation absorbed in my experiments by. the charcoal 
are independent of the humidity of the air carrying the 
emanation. 


Outer air bubbled through the solu- 


TOD... cip'arseusinnt'desanaee sasuldnn ean 


Outer air bubbled through the solu- 
tion, then dried by same calcium 
chloride as used with A ©) lecyesecue est 


_ Outer air bubbled through the solu- 
tion, then dried by same calcium 
chloride as used by A, then satu- 
rated by passing through 100 cm. 


caught. 


Emanation 


15 
it 


18:7 


Radium Emanation by Coconut Charcoal. 785 
EV 
Experiments with two charsoal tubes in series to find ont how 


the relative amounts of emanation absorbed by each of the 
tubes depend on the duration of the exposure. 


The silica tubes were connected up as follows :— 


Outer 
Radium Solution—1 metre of calcium chloride>Tube B —>Tube A>Gauge A>Pump. 
air 
TAsEe LED. 
Duration ; Amount of Amount of ‘ 
of Emanation Hmanation Ratio, B Means, 


Exposure. | caught by B. | caught by A. 


et 


——_—_—_ | ae 


3 hours. 50 2 04 
ee 48 efi “14 "10 
ae 49 6 "12 
53 hours. fia! AG 24 
saa 8:0 13 "16 - 
11 hours. 12:2 34 “28 28 
22 hours. 18°5 dist 6 * 
Stent 22°4 70 31 “31 
44 hours. 29°9 13:0 *45 
i 


| ik 349 12: "36 


* Neglected in finding the mean, 
Fig. 2. 


| 
TUBES IN SERIES | i i ie 


36 40 44 


16 20 24 28 
EXPOSURE (Hours) 
and the air sent through it at a constant rate (‘5 litre 
per minute). The radium solution used contained $ of 
3°14 x 10-° om: radium. 


Phil. Mag. 8. 6. Vol. 20. No. 118. Oct. 1910. 3 EF 


786 Mr. J. Satterly on the Absorption of 


The results are represented in fig. 2, and it is clearly 
shown that the charcoal in the first jaatie (B) approaches 
saturation when the exposure is long and the amount of 
emanation sent along the tube is high. 


“hs 
Percentage of Emanation absorbed by Coconut Charcoal. 


It is of interest to find what percentage of the emanation 
given off by a radium solution is absorbed by the charcoal 
under the conditions of my experiments. 

Using a solution containing 2 of 1°57 x 10-° om.* radium, 
S1X experiments were made with the silica tubes containing 
the coconut charcoal arranged as shown below :— 


Tube A or B—Gauge A 


oe pee 7 ee ; Sa 


\ Radium Solution—Tube B or A—Gauge BY Pump. 


The air-streams were adjusted to ‘48 litre per minute 
through each branch, and the exposure was continued for 
exactly 21 hours. Subtracting the amount of emanation 
caught in the tube in the “air alone” arm from the amount 
caught in the tube in the other arm, we get the amount 
caught by the charcoal from the solution. 

Before beginning this set of experiments the tubes had 
been emptied and made up afresh with 130 gms. of charcoal 
in each, so that the results are not absolutely comparable 
with earlier results. 

The results were :— 


ae Ao. ll VM ose UT Mean 4:8, 
Pues By rN tie oss Ye 2 5 ee 


the amounts being expressed in terms of the leaks produced 
in the testing vessel expressed in cms. per minute of my 
electrometer ‘scale, and reduced to a common sensitiveness 
(90 divisions per volt). 

It now remained to find the total amount of emanation 
produced by the same solution in 21 hours. To do this the 
bottle R containing the solution was connected up to a 
condenser © and heated in a brine bath (b.p. 105° C.), as 


* This was a portion of a fresh radium solution kindly supplied to the 
author by Professor Rutherford. 


Radium Emanation by Coconut Charcoal. 787 


sbown in the accompanying diagram (fig. 3). When the 
brine had been boiling for some time the clip T was opened, 


air bubbled through the solution, and the emanation drawn 
off from the solution and collected in the aspirator A and 
tested in the usual way. 

Table IV. gives the results, the leaks being reduced to the 
same electrometer sensitiveness as before. The last column 
is calculated from the second, by the help of tables similar 
to those given in my earlier papers *. 


TasBLe IV. 
Interval solution Emanation Hmanation 
had been resting generated in generated in 
since last heating. this interval. 21 hours. 
204 hours. tl 73 
742—Cs, 20°0 67 
a) Sot ior aE 7-4 
ee eee SI40 67 
Mean ...... 71 


The radium generates therefore an amount of emanation 
represented by 7-1 in 21 hours; and with an air-stream of 
"48 litre per minute the charcoal in (silica) tubes catches only 


* See also Kolowrat, Le Radium, 1909, pp, 193-5. 


788 Mr. A. Stephenson on Displacements 


4:5 (mean of A and B). The percentage caught is therefore 


~ 


ous or 63 per cent. (See also II. p. 783.) This is at the 


ordinary laboratory temperatures. 

Further experiments with air streams of magnitudes ‘11, 
*25, and ‘80 litre per minute showed that at these speeds the 
amounts of emanation caught were 86, 73, and 23 per cent. 
respectively. 


SUMMARY. 


Experiments have been made on the absorption of radium 
emanation by coconut charcoal, the emanation being carried 
to the charcoal by a stream of air. It has been found 


(a) that with weak solutions the amount of emanation 
absorbed in short exposures of the same time for the 
same strength of air-stream is proportional to the 
strength of the solution ; 

(b) that with the same solution and strength of air-stream 
the amount absorbed for exposures of different times 
does not increase in proportion to the time of expo- 
sure but falls off, showing that the charcoal is getting 
saturated ; 

(c) that under the conditions of the experiments the 
amount of emanation. absorbed does not depend on 
the humidity of the air ; 

(d) that with tubes 8 sq. cm. in cross-section containing a 
column 30 cms. long of coarsely powdered coconut- 
charcoal the amount of emanation absorbed when 
the air-stream is ‘5 litre per minute and the exposure 
is 21 hours, is only about 62 per cent. of the total 
amount of emanation carried by the air to the tube. 


In conclusion the author wishes to express his best thanks 
to Professor Sir J. J. Thomson for permission to carry out 
the above research at the Cavendish Laboratory. 


Cambridge, June 17, 1910. 


LXXXVIT. On Displacements in the Spectrum due to 
Pressure. By ANDREW STEPHENSON *. 
if has been suggested in connexion with the peculiar 
resonance effects exhibited in the spectra of sodium and 
other vapours, that a series of lines is, in certain cases, the 
spectroscopic analysis of the individual oscillatory motion of 
a single coordinate under the disturbing influence of some 


* Communicated by the Author. 


in the Spectrum due to Pressure. 789 


distinct normal motion. Under such influence, consisting 
in a periodic variation in the spring of the coordinate, the 
free motion changes from the simple oscillation 


A cos (ut +e) 


to the complex oscillation represented by 
S a, cos {\(utptrn)t +e}, 


where p and the ratios of the a’s are determined by the 
frequency and intensity of the disturbance. With regard to 
forced oscillations the coordinate absorbs energy from a 
direct force of frequency equal to that of any element in the 
series, storing it as a free (complex) oscillation—the property 
leading to the hypothesis. 

In the phenomena referred to n is small compared with p, 
and it is evident from the general method of analysis that 
even with a small variation in spring, the elementary ampli- 
tudes ...a_1, a, a... are comparable. The general solution 
may, however, be obtained by a very simple method :— 


2 = A(cosct+asin nt+e), 


where A and ¢ are arbitrary, is the solution of 


an? sin nt 


c+an cos nt “& + (c? + 4a7n? + 2acn cos nt + 4a?n? cos 2nt)e = 0, 


an equation determining the motion of a system of natural 
spring c+ 42?n”, subject to positional and motional forces 
the strengths of which are periodic functions of the time. 
n being small the periodic terms in n? are negligible com- 
pared with those in 7; thus 


z+ (p?+ 2aun cos nt)x = 0 


° 2 2,2 
eave w= Acos{(u-G*" )itasinnt+el, 
4p 

indicating a reduction in frequency of the second order of 
small quantities. The square of the amplitude of spring 
variation is proportional to the energy of the normal motion 
producing it ; and such motion being subject to dissipation 
according to the exponential law, for a steady state its 
energy must be proportional to the pressure—the temperature 
being assumed constant. Thus the reduction in the frequency 
of the series is directly proportional to the pressure. 

Although the equation of motion cin always be reduced to 


790 Geological Society :— 


the above form, the quantity directly affected when the dis- 
turbance is slow, is the reciprocal of the spring, which in 
general undergoes (i.) achange in its mean value proportional 
to the pressure, and (ii.) a periodic variation: the latter gives 
an increase in frequency, the former either an increase or 
diminution. Thus we can assert only that there is a change 
of frequency proportional to the pressure. 

Reductions proportional to the pressure have been observed 
in the case of iron vapour *, but the hypothesis does not 
appear of marked value in this instance owing to the absence 
of specially selective effects. Evidently the statical influence 
of the pressure is predominant. 

In the case of sodium many resonance series appear to 
have the same frequency difference. From the present 
standpoint it seems probable, therefore, that the vapour 
would strongly absorb long waves of this frequency, n/27, 
and show selective refraction in its neighbourhood. Such 
absorption would be accompanied by displacements of the 
series towards the violet end of the specirum under the 
influence of the intensified periodic variation in the reciprocal 
of the spring. 

It is hardly necessary to point out that slow normal 
oscillations within the vibrating system are sufficient to 
account for the complexity of linear spectra, and the varia- 
tions in relative brightness at different pressures. 


July 1910. 


LXXXVIII. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from p. 544. ] 


February 23rd, 1910.—Prof. W. W. Watts, Sc.D., M.Sc., F.RB.S., 
President, in the Chair. ' 


HE following communication was read :— 


‘Metamorphism around the Ross of Mull Granite’ By 
Thomas Owen Bosworth, B.A., B.Sc., F.G.S. 


The Ross of Mull granite is a coarsely crystalline plutonic mass, 
forming the western portion of the Ross of Mull and extending 
over some 20 square miles, 

The intrusion is conspicuously later than the Moine rocks, and 


* W.G. Duffield, “ On the Effect of Pressure on the Arc Spectrum of 
Iron,” Phil. Trans. 1907, 


Metamorphism around the Ross of Mull Granite. 791 


is regarded as one of the ‘ newer granites’. The rock shows very 
little evidence of faulting or movement of any kind, and is traversed 
by sheets of mica-trap. The eastern boundary of the granite is a 
very intricate line of junction with typical Moine Schists and 
Gneisses, into which it has been intruded.  Injection-breccias 
occur along the margin, where the granite is crowded with schist- 
inclusions. 

The changes in the pelitic schists are of two kinds, and are con- 
sidered under separate headings (a) and (bd) below. 

(a) Impregnation.—The schists have been impregnated with 
the granite in a very intimate manner:—(1) Along irregular 
eracks; (2) Along bedding-planes; (8) Along strain-slip; and 
(4) Along foliation. 

Variously banded rocks have been thus produced, which suggest 
how readily these processes, carried out on a large scale, would 
convert pelitic sediments from the state of schists into crystalline 
igneous gneisses. 

(6) Thermal Metamorphism.—In some places the pelitic 
gneiss in contact with the granite, and commonly the masses in- 
cluded in the granite, have been very highly altered. The new 
minerals formed are sillimanite, andalusite, cordierite, and green 
spinel ; and these are present in such amount that their formation 
must have been accompanied by much recrystallization among ee 
quartz, felspar, and mica also. 

Sillimanite is the most abundant new mineral, and occurs not 
only as fibrolite throughout the rock, but also in larger crystals 
which are often grouped together in prismatic aggregates. These 
aggregates weather out as conspicuous knobs, measuring about an 
inch across. 

Under the microscope, the sillimanite is seen to enclose large 
numbers of grains of green spinel. The cross-sections of silli- 
manite are diamond-shaped, and show a pinacoidal cleavage; their 
colour between crossed nicols is a very low grey, and good inter- 
ference-figures are obtained. 

The association of minerals in the schists is the same as that 
noticed at the margin of the Ben Cruachan ‘ newer granite ’-mass, 
and also at the margin of ‘newer granite’ at Netherly in Elgin. 

Tourmaline, kyanite, and staurolite also occur in the Moine 
Schists of Mull, but are in no way connected with the granite. 


March 9th.—Prof. W. W. Watts, Sc.D., M.Sc., F.R.S., 
President, in the Chair. 


The following communication was read :— 


‘The Carboniferous Succession in Gower (Glamorgazshire).’ By 
Ernest Edward Leslie Dixon, B.Sc., F.G.S., and Arthur Vaughan, 
BUA... D.Se., EGS: 


The succession in three districts in Gower is described, the 


792 Geological Society. 


districts being so situated that a comparison of their respective 
developments can be interpreted in the light of the fact that, during 
Avonian time, the nearest coast lay to the north, with a general 
east-and-west trend. With the description of the lithological sequence 
are included notes on some breccia-like limestones, characteristic 
of D, and on ‘lagoon-phases’ and the origin of radiolarian cherts. 
To the faunal lists are added notes on the D,—D, phase of the 
Dibunophyllum Zone, which distinguishes Gower from the rest 
of the South-Western Province at present known, and on the 
correlation of that zone with the Upper Bernician of Northumber- 
land. From the faunal sequence it is concluded that the zones 
Z, C, 8, D, and D, (the K Zone is poorly exposed) are characterized 
by the same assemblages as in the Bristol area. 

The lithological sequence shows (1) that over the whole area the 
depth of the Carboniferous sea underwent a complete cycle of 
intermittent change during Lower Avonian time, the initial deepen- 
ing being followed by gradual shallowing up to the top of the 
lower part, C,, of the Syringothyris Zone, which was deposited 
almost at sea-level; (2) that a similar cycle marked the ensuing 
period up to the top of the Seminula Zone; (3) that a similar but 
smaller cycle took place in the Dibunophyllum Zone, the latter 
actually reaching the surface; and (4) that a fourth cycle, com- 
mencing with a far-reaching physiographic change, characterized 
the Posidonomya Zone. 

Further, a comparison of the sequences and thicknesses in the 
three districts shows that, not only were the downward movements 
of the sea-bottom during the first two cycles greater in the south 
than in the north, but also that the axis on which the movement 
during the first cycle hinged was different in direction from the 
axis during the second cycle. The bearing of these movements on 
the question of the delimitation of the divisions of the Avonian is 
then discussed. They suggest that the base of the upper part, C,, 
of the Syringothyris Zone should form the base of the Upper 
Avonian. On the other hand, the base of C, in at least two 
localities is closely connected, faunally, with the zones below, 
whereas the fauna of the main mass of C, passes into 8, without 
appreciable change other than the introduction of Lithostrotion. 
It will, therefore, in all probability be decided that the break 
between the Lower and the Upper Avonian should be taken at a level 
within C, rather than at the base of the SeminuJa Zone. For the 
present, however, this question must be deferred, since it concerns 
the whole extent of the formation in Belgium, the North of France, 
and the British Isles. 

The paper concludes with notes on some of the corals and brachio- 
pods, including one new species of coral and two new species and 
a new variety of brachiopod. 


JONES & ROBERTS, Phil, Mag. Ser. 6, Vol. 20, Pl. XIII. 


The plates belonging to 
here but not in their 


as 2 04 - N,— nN, = 200°6. 


Fi4G. 2. 


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Rigs 3S: 


Woop, Phil, Mag. Ser. 6, Vol. 20, Pl. XIV. 


Fic. c Pies 2. 


Fig. 5. 


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Phil. Mag. Ser, 6, Vol. 20, Pl. XVI. 


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woop. Phil. Mag. Ser. 6, Vol. 20, Pl. XVII. 


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Phil. Mag. Ser. 6, Vol. 20, Pl. XII. 


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Phil. Mag. Ser. 6, Vol. 20, Pl. XII. 


BHRRY, 


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Phil. Mag. Ser. 6, Vol. 20, Pl. XV. 


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GARRETT. 


CURRENT 


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THE 


LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


NOVEMBER 1910. 


LXXXIX. On the Critical Phenomena of Ether: 
By ¥. B. Youne, B.A., B.Sc.* 


a has been observed by numerous investigators that when 

a liquid in contact with its vapour is heated to the 
critical temperature of vaporization, the meniscus disappears 
whilst the density of the liquid is still distinctly greater than 
that of the vapour, and that this difference of density may 
persist for an appreciable time even if the temperature is 
still further raised. According to the simple theory of 
- continuity of state, the liquid and vapour should become 
identical at the critical temperature +. Various explanations 
of this phenomenon have been offered, some of which are 
intended to reconcile it with Andrews’s theory, whilst others 
utilize it as evidence of the insufficiency of that theory. The 
chief views which have been presented are the following :— 


A. Explanations involving no modification of Andrews’s 
Theory. 

G.) Gouy (1) points out that owing to the influence of 
gravity the pressure in the substance is not uniform 
but increases from the top downwards. Since the 
substance when exactly at its critical state is quite 
abnormally compressible, an appreciable discontinuity 
may be produced in the density of the substance at 
that level at which the pressure is equal to the critical 
pressure. 


* Communicated by Prof. A. P. Chattock. 

+ S. Young has shown this to be true in the case of normal pentane 
(Trans. Chem. Soc. vol. lxxi, p. 446, 1897). 

Piz. Mag. SvGo Nov. 20. No. 119. Nowe 1910), ° 3G: 


794 Mr. F. B. Young on the 


(ii.) Villard (2) points out that at the critical temperature 
and just above it, the dilatation with rise of tempe- 
rature is extremely great. He attributes the differ- 
ence of density to a slight but persistent difference in 
temperature. In support of his explanation he has 
reproduced the difference of density in a tube of 
ethylene by gently heating the upper portion. Since 
the mean specific volume was greater than the critical 
volume, the effect must have been quite distinct from 
that predicted by Gouy. 

(iii.) Onnes and Verschaftelt (3) ascribe the phenomenon 
largely to the presence of traces of impurity in the 
substance. If the substance contain a slight ad- 
mixture of some other more volatile substance, the 
latter might tend to collect in the vapour phase and, 
by its partial pressure, would produce an effect similar 
to that ascribed by Gouy to gravity. 


B. Liquidogenic Hypotheses. 


(iv.) Some investigators (amongst whom are De Heen 
and Traube) * consider that the difference of density 
corresponds to an actual difference of molecular 
structure. The liquid molecules or liguidons are of 
greater mass than the gaseous molecules or gasons ; 
the meniscus disappears when the two phases become 
mutually soluble in all proportions, but homogeneity 
of molecular structure occurs only at a temperature 
(T.), which is higher than that of the disappearance of 
the meniscus (@,). It is urged that the differences of 
density observed are both too great and too persistent. 
to be due to differences either of pressure or of tempe- 
rature, whilst it is claimed that impurities are elimi- 
nated by the methods of preparation. 


The investigation to be described was undertaken with the 
purpose of examining the Cagniard-Latour phenomena in 
the light of the various explanations offered: 


* The results obtained by De Heen (Mathias, Point Critique des 
Corps Purs, p. 197, P. de Heen, Bull. de l’ Acad. roy. de Belgique |3}| 
t. xxxi. pp. 147 et 379, 1896), and by Teichner (Ann. d. Physik, Feb. 
1904) have been critically examined in the laboratory of K. Onnes (doe. 
cit.), and the results of the inquiries warrant a strong presumption that 
the marked differences of density are to be attributed to the presence of 
a small percentage of impurity in the substance assumed to be pure. 
Since, however, Mathias regards the controversy between the Classical 
and Liquidogenic Theories as still an open one, additional experimental 
material relating to the critical phenomena of pure substances seemed. 
to be highly desirable. : 


Critical Phenomena of Ether. 795 


Elaborate precautions were taken to secure the purity of 
the ether, and especially to free it from the last traces of 
permanent gas. The submersed bell described in Section A. 
proved a delicate means of detecting the slightest traces of 
gaseous impurity, and showed the precautions taken to be 
quite necessary for the attainment of the best results. It 
was possible thereby not only to test the efficiency of the 
means adopted for purifying the ether, but also to classify 
the tubes prepared according to the relative amounts of 
impurity contained. The observations made with these 
tubes proved that extremely slight traces of gaseous impurity 
might visibly modify the critical phenomena. | 

By means of the bell above mentioned the part played by 
hydrostatic pressure in the Cagniard-Latour phenomena was 
tested, and found to be probably negligible. 

The effect produced by Villard with ethylene was repro- 
duced in the tubes of ether, and the density difference so 
produced was carefully compared with that associated with 
the Cagniard-Latour phenomena in the pure substance. The 
conclusion arrived at was that these were essentially 
identical. | 

The opalescent phenomena were carefully examined, and 
explanations are suggested of some of the peculiar opalescent 
effects described by earlier investigators. Some account is 
also given of preliminary observations which were made 
with a view to determining the nature of the opalescent 
particles. 

The investigation has led to the general conclusion that 
the phenomena observed in the pure substance may readily 
be explained in accordance with the theory of Andrews, 
whilst the phenomena, both qualitative and quantitative, 
which have been advanced as demonstrating most strongly 
the insufficiency of that theory are to be observed only in 
tubes which contain distinct traces of impurity. 


Before proceeding to the experimental work, I would like 
to point out that if the differences of density can be 
adequately accounted for, the various other phenomena 
which have been heid to invalidate the Classical Theory 
need present no difficulty. Cailletet and Collardeau (4) 
have shown that iodine dissolved in liquid CO, remains in 
solution in the denser substance above @,, although the less 
dense substance does not acquire the property of dissolving 
iodine. Hagenbach (5) found that the resistance of a very 
dilute solution of sodium iodide or bromide in sulphur 
dioxide remained less in the one portion of the tube than 

3G 2 : 


796 Mr. F. B. Young on the 


in the lower for two or three hours after the temperature 
had been raised above the Cagniard-Latour temperature. 
Eversheim (6) observed similar effects in ammonia and SOx, 
whilst the dielectric constant for HS differed also in similar 
fashion. If, however, the density differences can be explained 
in accordance with Andrews’s theory these latter forms of 
evidence become inconclusive. For though, by definition, 
the liquid and vapour cease to exist above the Cagniard- 
Latour temperature, there is no cogent reason why their 
peculiar properties should forthwith disappear. The diagram 
of Andrews is apt to give an impression of discontinuity at 
6, which in practice is nonexistent. The ordinate AB (fig. 1), 


for which v is equal to the critical volume V,, appears in 
meeting the saturation curve to strike a point of sudden 
transition ; this is because of the impossibility of realizing 
the theoretical isothermals within that curve. In practice, 
however, as the explanations of both Gouy and Villard 
indicate, the bulk of the substance does not, during rise of 
temperature, pass through @, along the isothermal AB, but, 
roughly speaking, along two ordinates DC and EF, for 
which v << V, and v> V, respectively. In this case the dis- 
continuity at @, ceases to be apparent. Hence it is not 
unreasonable to assume that the continued difference of 
density will be accompanied by a continued difference of 
properties; and if adequate causes for the observed persistency 
of the density difference can be found which are in accord- 
ance with Andrews’s theory, the same causes may be held 
to produce indirectly the observed differences of electrical 
and chemical properties. 


Critical Phenomena oj Ether. 797 


A. Preparation of the Tubes. 


The tubes employed were of the design shown in figure 2. 
The Natterer tube AD is of Jena glass tubing about 1 mm. 
in thickness, about 8 or 10 mm. in internal diameter, 
and from 13 to 16 cm. in length. Inside AD slides 
freely a piece of Jena-glass tubing C about 3 cm. in 
length, which is closed at the upper end and thus 
forms a bell. The bell-tube is fused to a similar 
piece of tubing B in which are hermetically sealed 
some tightly packed iron filings; by means of a 
sufficiently powerful electromagnet, C and B may be 
raised at will. The ether is introduced through the 
constricted end D which is afterwards sealed off. 

The special purpose of this design is explained 
later, but, as has already been stated, the bell proved 
to be a very efficient detector of gaseous impurities, 
If such impurity is present, it is to be found chiefly 
in the vapour. By inverting the tube and then 
slowly bringing it back to the position shown in 
fig. 2, the bell-tube may be made to fall into the 
liquid, carrying within it a portion of the vapour. 
If the vapour is pure it readily condenses, but if it 
is mixed with gas a bubble remains which dissolves 
very slowly, since it is subjected to a pressure of 
only a few centimetres of liquid ether. As a further 
test, C may be filled by ebullition. If the end D is 
gently heated, the ether boils briskly, whilst reflux 
condensation occurs at end A. By this means the 
gaseous impurity, if present, is brought mainly to 
the top of the tube, and C is filled with compa- 
ratively pure vapour from beneath. This sample of 
vapour may then be compared with that previously 
obtained. 

In filling the earlier tubes various methods of 
removing the air were tried. In some cases, a large 
excess of ether was introduced and the surplus was 
pumped off with a Tépler pump. In other cases, 
the method described by Travers and Usher (7) was 
adopted : the ether was frozen with liquid air whilst the tube 
was exhausted, and the excess of ether was then pumped off 
before sealing the tube with the blowpipe. In all cases, 
however, very distinct traces of gaseous impurity were ob- 
servable: the vapour brought down from end A was far more 
persistent than that produced in the bell-tube by ebullition, 
and in order to secure its condensation it was necessary to 


Fig. 2. 


798 Mr. F. B. Young on the 


raise the pressure by gently heating the upper part of the 
tube. 

Since the removal of gaseous impurity was of the utmost 
importance, the methods of filling were carefully examined. 
Tube XII. was filled with excess of ether and connected 
through a prolongation of the capillary D (fig. 2) and a 
short piece of rubber pressure-tubing to a Topler pump. 
The ether having been frozen with liquid air, the tube was 
exhausted. When the ether was allowed to thaw a brisk 
effervescence of bubbles proceeded from the surface of the 
dwindling solid lump. This was the dissolved air which had 
separated out when the ether was frozen. Not all the air, 
however, escaped in this way. After a little of the ether 
had vaporized, the freezing and thawing were repeated and 
again bubbles were evolved, though fewer than _ before. 
Though the ether was frozen four times, a few bubbles rose 
after the final thawing. (In the case of another tube bubbles 
were evolved after the ninth freezing.) The trouble was 
largely due to the partial solution of the bubbles as they 
rose, for some of the smaller bubbles visibly dwindled in 
size as they ascended. ‘The tube was finally sealed off at the 
constriction whilst the ether vapour was passing freely into 
the pump. 

For purposes of comparison another tube A was filled with 
ether. The constricted end had been drawn out and hent so 
that the tip might be immersed in mercury. About two- 
thirds of the ether was boiled off and the tube was then 
sealed. Tubes XII. and A were then immersed in liquid 
air : when the ether thawed, an eye estimation of the bubbles 
evolved led to the conclusion that tube A contained rather 
more permanent gas than tube XII. 

The importance of reducing the pressure to a minimum, 
if the freezing process is employed, was shown by freezing 
and thawing ether at atmospheric pressure. Though 
numerous bubbles appeared upon the surface of the solid lump 
as it melted away, scarcely one left the surface, so rapidly 
did they dissolve. 

A further test showed the extreme solubility of air in 
ether at the atmospheric pressure. A tube which had been 
nearly freed from permanent gas in the manner above 
explained was detached from the pump, so that the ether 
was again exposed to the atmosphere. After a few moments 
it was replaced on the pump, care being taken to avoid 
shaking the tube more than was necessary. On freezing 
the ether and thawing it under reduced pressure, the effer- 
vescence was not perceptibly less than it had been originally. 

These tests suggest that the test commonly relied upon for 


ii i i i ii i — 


Critical Phenomena of Ether. 199 


judging the absence of permanent gas—namely, the absence 
of bubbles in a capillary tube containing the ether—cannot 
be relied upon if the ether is under a pressure approaching 
that of the atmosphere. 

In some of the earlier tubes, the traces of gaseous impurity 
were more distinct after the tubes had been raised to a high 
temperature than they were immediately after the process of 
filling. Since the ether which was used in these tubes had 
been kept for some months in a stoppered bottle placed in 
a desiccator, it is possible that it had been attacked by the 
atmospheric oxygen with the formation of products which 
decomposed at a high temperature. In filiing the later 
tubes, the ether to be used was finally distilled over potassium 
and sodium alloy within an hour or two of the process of 
filling. 

Fig. 3. 


The following rather drastic method of filling the tubes 
was finally tound necessary. The tubes to be filled (A, B,C, 
and D in fig. 3) were fused to a horizontal tube H, from 


800 Mr. F. B. Young on the 


which were suspended also the two reservoirs F and G and 
the bulb of charcoal H. One end of tube E terminated in 
the fine capillary tube J, the sealed tip of which was drawn 
out very thin and bent, so that it might readily be snapped 
by a gentle pressure. The other end of EH communicated 
with a Tépler pump through the mercury trap K, the arms 
of which were about 90 em. in height. Between K and the 
pump was interposed the trap M: this was immersed in 
liquid air, and by arresting the ether vapour prevented the 
fouling and consequent loss of efficiency of the pump. Since 
the experimental tubes were of Jena glass it was necessary 
to construct the whole of the filling apparatus of the same 
material. The connexion with the pump was made by means 
of the mercury-sealed ground joint N. 

In order to free the tubes from moisture and adsorbed air, 
the apparatus from H to G was covered with asbestos mill- 
board and heated over a tube furnace. The tubes were 
evacuated by means of the pump, and the trap K was left 
open so that they were in communication with the phos- 
phorus-pentoxide tube attached to the pump. This process 
was repeated several times. Just before carrying out the 
filling operations the trap K was closed ; the charcoal bulb 
was immersed in liquid air for about an hour and was then 
fused off, so that a Dewar vacuum was created in the tubes. 

The apparatus from K to G was heated well above the 
boiling-point of ether. The capillary tube J was then intro- 
duced into the freshly prepared ether, and its tip was broken 
by pressure against the bottom of the bottle. The first 
portion of ether which ascended the capillary vaporized, 
until the internal pressure was equal to that of the atmo- 
sphere. The reservoir G was then cooled with water, the 
remainder of the apparatus being kept hotas before. When 
the desired amount of ether (about 45 c.c.) had collected in 
G, the flow was checked by immersing G in water at about 
35° C., and the capillary tube was sealed off near the top. 

The ether in G was then frozen with liquid air. The 
mercury in K, acting as a manometer, still showed a pressure 
in the tubes of 8°8 cm., this being almost entirely due to the 
air which had entered in solution. M having been sur- 
rounded with liquid air, K was opened and the air was 
pumped off. The ether was then allowed to thaw, and a 
first portion passed off as vapour through K. The bulk of 
the ether was condensed in F by means of liquid air; a little 
of the remainder was then condensed in the experimental 
tubes, after which the ether in F was allowed to thaw and 
the first portion of it passed off through K together with the 


— ae 


Critical Phenomena of Ether. 801 


ether in the experimental tubes and the remainder in G. 
The ether was distilled to and fro between F and G, the 
operations above described being repeated at each reversal. 
In this way the first and last fractions were each time 
removed, and the experimental tubes were flushed out with 
ether vapour. K was so manipulated as to prevent the 
return of any vapour from the pump. The distillation was 
accelerated, when necessary, by dipping the reservoir in a 
beaker of methylated spirit, and the spirit served the further 
useful purpose of keeping the reservoir free from frost, so 
that careful observation was possible. The bubbles evolved 
on thawing decreased at each successive distillation till at 
the commencement of the sixth distillation none were visible. 
The sixth distillate was then frozen and allowed to thaw, 
whilst one of the empty tubes was immersed in liquid air in 
order that the ether might thaw under a minimum pressure. 
As there was again an entire absence of bubbles, the process 
of the purification was considered complete. Hach experi- 
mental tube in turn was then filled by condensation with a 
slight excess of ether ; the excess was distilled off, and the 
tube sealed off at the constriction with the blowpipe. 

Some of the tubes were sealed whilst the vapour was 
issuing freely ; others were immersed, before sealing, in 
liquid air, so that the constriction was free from vapour. 
The latter tubes showed a slight but distinct superiority over 
the former when tested for the presence of permanent gas. 
From this it appears that the vapour in contact with the hot 
glass undergoes decomposition. 

For the above process, Kahlbaum’s ether distilled over 
sodium was used after further treatment. About a third of 
a litre of the ether was placed in a well-stoppered bottle with 
some sodium and potassium alloy. The bottle was wrapped 
in opaque paper and placed in a mechanical shaker, by means 
of which it was repeatedly shaken for several hours at a 
time. ‘This plan was adopted in view of a suggestion that 
the efficiency of the solid metal is quickly impaired by the 
oxidation of the surface: by means of the shaking the surface 
of the liquid alloy was constantly renewed. After about 
three months of this treatment the ether was fractionated 
over the alloy three times through a five-section Young 
fractionating column. The ether was protected from atmo- 
spheric moisture by a guard-tube of phosphorus pentoxide, 
the receiver ,being ground to the condenser tube; and by 
introducing the liquid into the apparatus in the manner 
previously described, the necessity for pouring it through 
the air was avoided. The thermometer employed was 


802 Mr. F. B. Young on the 


graduated in tenths of a degree and was calibrated by 
means of a similar Reichsanstalt thermometer. The boiling- 
point of the selected fraction was 34°53-34°°54 ©. (N.T.P. 
at lat. 45°). 

The B.P. is lower than the generally accepted value 
(34°°6 C.); but, according to Wade and Finnemore (8), 
whose work on the purification of ether was unfortunately 
not published until after the above preparation had been 
completed, the boiling-point of pure ether is very near 
34°50 C., and the ether prepared was therefore not really 
freed from the last traces of alcohol. This possibility is 
remembered in discussing the experimental results obtained. 

The following are details of the various tubes to which 
reference has been or will be made in this paper :— 

Tube VII. was filled with ether kindly supplied by 
Mr. Usher and prepared in the manner described by Travers 
and Usher (7); it had, however, been kept for some time. 
The tube was filled with the aid of liquid air and the simple 
Ldpler pump, the ether being frozen seven times ; it was 
sealed with the vapour issuing. 

Tube XII. was filled with ether freshly prepared by the 
method of Travers and Usher (7), sodium and potassium 
alloy, however, being substituted for potassium wire. It was 
filled in the same way as tube VIL., the ether being frozen four 
times. 

Tube XIII. was filled with Kahlbaum’s ether treated with 
sodium and potassium alloy. It was filled by means of the 
special apparatus described and sealed whilst the vapour was 
issuing. 

Tube XIV. was filled in the same manner and at the same 
time as tube XIII., but was sealed whilst the ether was frozen. 

For purposes of comparison, tubes VII., XII., XIIL., and 
XIV. were fastened side by side and immersed in a tank of 
water at a temperature of about 10° C. After immersion for 
about 15 minutes, the bell-tubes were filled simultaneously 
by inversion of the experimental tubes, and observations 
were made of the condensation of the vapour carried down 
by the bells. Tubes XIII. and XIV. were markedly superior 
to tubes VII. and XII, whilst XIV. was distinctly freer 
from permanent gas than XIII. The observations are tabu- 
lated in Table I. 

It may be mentioned that the head of liquid under which 
the vapour condensed was roughly the same except in the 
case of tube XIT., in which it was about one-half as great as in 
the other tubes. Hence the apparent similarity of tubes VII. 
and XII. implies that tube XII. is somewhat superior to 


Critical Phenomena of Ether. 803 


tube VII. In every case the bubbles could be made to vanish 
by gently warming the upper layers of ether and thus adding 
the vapour pressure to the hydrostatic pressure, so that had 
the vapour been tested under atmospheric pressure it would 
probably have completely condensed in every case. 


TaBLe I. 
: 
| | Observations of condensation 
| of vapour. Fraction of | Critical Tem- 
| No.of tube occupied | perature (of 
| Tube by liquid reappearance 
Bell filled Bell filled at U° CO. of meniscus). 
by inversion. | by ebullition. 
iS a 2yremained | Vapour con- 0°335 193°°59 OC. 
after 13 mts. densed in 
(2 remained 1 mt. 
after 90 mts.) 
; XII. ...... Zyremained | Vapour con- _ ~- 
after 15 mts. densed in 
22 mts. 
1b ER yo remained Vapour con- 0°385 193° 58 C. 
after 13 mts. | densed in 
(#5 remained 235 mts. 
after 90 mts.) | 
BEY 52 Sons Condensation | Vapour con- 0:345 193°°59 O. 
complete in | densed in 
13 mts. 223 mts. 
The bell-tubes were then filled by ebullition. The bells 


having been filled with liquid, the experimental tubes were 
placed point downwards over a hot-water coil for five minutes. 
They were then plunged, still pomt downwards, into the 
water-tank. In this case the vapour in tube VII. condensed 
more readily than that in the remaining tubes, the conden- 
sation being probably assisted by the partial pressure of the 
gaseous impurity. The distinctly greater persistence of the 
vapour in tube XIV. may be accounted for by the additional 
time required for the preliminary establishment of equilibrium 
of temperature. The time required for the condensation of 
the vapour in tube XIY. does not seem unduly great when it 
is considered that to maintain the two menisci in equilibrium 
at the average difference of level (about 6 cm.) a difference 
in temperature of only 0°2 C. between the menisci was 
necessary, or a difference in temperature of 0°1 C. between 
either meniscus and the water-bath. During the process 


804 Mr. F. B. Young on the 


about 0°1 of a calorie was absorbed at the upper meniscus 
and evolved at the lower ; the dissipation of heat from the 
bell, moreover, occurred through two thicknesses of glass 
separated by a thin film of liquid. 

The tests have only a rough quantitative value since the 
tubes were not identical in their dimensions. They suffice to 
show, however, that (a) the ether should be prepared shortly 
before being placed in the tubes; (b) the last traces of 
gaseous impurity can be removed only with the utmost diffi- 
culty ; (c) if the tubes are to be sealed with the blowpipe 
the pressure of the ether vapour should previously be reduced 
toa minimum. Lach of the above experiments was checked 
by duplicate tubes. 

In Table I. are included for reference the proportions 
of the tubes filled by the liquid at 0° C. and the critical 
temperature of the ether in tubes VII., XIII., and XIV. 
(Lube XII. exploded before observations could be made.) 

The temperature given is that at which the meniscus was 
first perceived as the ether was very slowly cooled. Since 
in neither of the tubes was the mean density exactly equal 
to the critical density, the actual Cagniard-Latour tempe- 
rature (@,) is probably slightly higher than that given. The 
critical temperature for tube XIV. is the mean of seven 
observations, and the critical temperatures for tubes VII. 
and XIII. are obtained by direct comparison with tube XIV,, 
the tubes being observed simultaneously. 


B. LNisposition of Apparatus. 


The tube under observation was supported on a glass rod 
in the vapour jacket E (fig. 4). This jacket was surrounded 
by two guard-tubes forming a double air jacket, whilst an 
inner tube F, which was widened at the bottom to fit the 
vapour jacket rather closely, still further protected the tube 
from temperature disturbances. A coil of metal tube H acted 
as a condenser, a stream of cold water being passed through 
it. The tin cone K deflected any drops of condensed liquid 
to the sides of the vapour jacket. The asbestos cone G 
protected the vapour from the heat of the flame which 
was placed beneath. The vapour jacket was placed 
between the poles L of a powerful electromagnet which 
was mounted on a small counterpoised lift so that it could 
be readily raised or lowered. By means of the magnet 
the bell C (fig. 2) could be raised at will. The upper portion 
of the experimental tube was in some cases surrounded by an 
open coil of fine iron wire N having a resistance of about 
2 ohms, the ends of which were connected through the needles 
M to electrical terminals. ‘The vapour jacket was connected, 


Critical Phenomena of Ether. 805 


through the side tube, with an arrangement for maintaining 
constant the internal pressure (v. Appendix). 

The liquid used in the vapour jacket was aniline. For this 
purpose Kahlbaum’s Aniline was redistilled and the middle 
fraction was employed. The temperatures were obtained 
from the vapour-pressure readings given by Travers and 
Usher (7). In the few cases in which these readings were 
exceeded, it was considered sufficient to extrapolate the given 


Fig. 4. 


——$— 


i 
: 
y 
v 
4 
nN 


S 


values, since in those cases an approximate temperature 
sufficed for the purpose. 

The tube F was sufficiently wide to accommodate two 
experimental tubes side by side, and where comparisons are 
made between two tubes, the tubes were heated simultaneously, 
so that the conditions of temperature were identical. 


806 Mr. F. B. Young on the 


C. General Phenomena. 


A brief statement is given of the general phenomena 
observed in tube XIV. in the neighbourhood of the critical 
temperature. 

(1) If the tube was slowly cooled after being maintained 
at a temperature about 3° C. above @, :— 

(a) A faint opalescent haze appeared throughout the tube 
and gradually became denser, at first very slowly, but 
more rapidly as the temperature @, was approached. 
Just above that temperature a dense opaque fog 
pervaded the tube and the substance appeared greatly 
agitated. 

(b) At 6, a faint, flat meniscus was perceived slowly rising 
around the plunger B (fig. 2). A brisk ebullition of 
fine bubbles was generally discernible through the 
dense opalescence which still persisted after the 
appearance of the meniscus. 

(c) Asthe temperature was further reduced the opalescence 
became fainter and at length disappeared. The ebul- 
lition became more marked, while condensation was 
apparent in the vapour. The meniscus gradually rose, 
becoming at the same time clearer and assuming a 
concave shape. (If the reduction of temperature was 
effected in sudden drops of about 0°02 the tube 
became filled each time with an oily cloud which 
dissolved into a rain of bubbles and drops rising and 
falling respectively into the meniscus.) 


These appearances may be regarded as belonging to con- 
ditions of equilibrium, since by arresting the fall of tempe- 
rature the changes described could be arrested at any point. 

(2) If the temperature was raised extremely slowly 
(e.g. 0°01 C. in 5 mts.) it was possible to approximate to a 
reversal of the phenomena of slow cooling, the condensation 
phenomena in the vapour phase excepted. It was difficult, 
however, to avoid the effects of lag detailed in (3) and (4). 

(3) If the temperature was raised as quickly as possible to 
the neighbourhood of @,, the liquid quickly became opalescent 
and after a few minutes the vapour also became opalescent. 
At the same time the meniscus fell and, given sufficient time, 
fell below the top of the plunger, bey yond which point its 
movements were difficult to follow. 

(4) If the temperature was rapidly eat a little above 
@. (say 0°:2 C.) and then maintained constant, the meniscus 
quickly became indefinite in outline and nebulous in appear- 
ance; it assumed at the same time a slightly convex 


Critical Phenomena of Ether. 807 


appearance (v. Section E). The substance beneath the 
disappearing meniscus quickly became densely opalescent ; 
the top of the opalescent column slowly became less defined, 
and the substance above it also grew opalescent till the 
opalescence was uniform. The higher the temperature was 
the more rapidly did the meniscus fade and the less marked 
were the opalescent phenomena. In each case the final 
appearance was that produced by slowly cooling the tube as 
in 1 (a) to the same temperature. The height at which the 
meniscus faded varied considerably with the rapidity with 
which the temperature was raised to the desired temperature. 
The more rapidly the temperature was raised the greater was 
this height, which might be as much as 3°5 cm. above the 
plunger. 

(5) If the temperature was quickly raised as in (4) and 
was then allowed to fall steadily before the liquid phase had 
become opalescent, an opalescent zone gradually appeared at 
the level at which the meniscus was fading and became denser 
as 0, was approached, till at length a meniscus appeared within 
the zone and the opalescence then faded away. The pheno- 
mena were practically a rapid repetition of those described 
in 1 (a) (0) and (¢), confined, however, to a very short length 
of tube. The more promptly these operations were carried 
out, the shallower and more definite was the zone, the upper 
and lower limits of which sometimes had almost the sharpness 
of a meniscus. 

In tubes VIL. and XIII. the equilibrium phenomena were 
similar to those of tube XIV. except that in tube XIII. the 
meniscus was higher, the more nearly the temperature 
approximated to 6,. The variations in the transitory pheno- 
mena are dealt with in later sections. 


D. Influence of Gravity. 


The tube described in Section A was designed for the 
purpose of testing the extent to which the abnormal com- 
pressibility at the critical volume predicted by Gouy trom 
Clausius’s equation accounted for the observed phenomena. 
The intention was to raise the bell to such a height that the 
meniscus disappeared within it, giving place to a zone of 
transition. The bell would then be gently lowered, carrying 
with it the transition zone. In this way the pressure at that. 
zone would be increased by the hydrostatic pressure of the 
column of ether through which it was lowered, and theoreti- 
cally the zone should then rise in the bell. . 

It was essential that the last traces of permanent gas. 


808 Mr. I’. B. Young on the 


should be removed. The earlier tubes distinctly showed 
effects due to the presence of such traces. The observations 
made with tube VII. may be taken as typical. When the 
vapour jacket was first heated, the aniline vapour, as it rose 
round the tube, produced a reflux condensation which, as 
before explained, had the effect of collecting the bulk of the 
gaseous impurity at the top of the tube, whilst the bell was 
filled with comparatively pure vapour. The tube was then 
maintained at a constant temperature of about 185° C. The 
meniscus was observed to rise slowly in the bell; after 
7 minutes it had risen about halfway, and after 20 minutes 
about three-quarters of the way up the bell. The bell was 
next raised above the liquid so that it was filled with vapour 
from above. It was then lowered to the bottom, and the 
meniscus was again observed. The meniscus now remained 
at the mouth of the bell and had not risen at all at the 
expiration of 30 minutes. 

Tube XIV. was heated simultaneously with tube VIL., and 
in both of the above tests the meniscus remained at the 
mouth of the bell. It is evident, therefore, that the meniscus 
in tube VII. had in the first case been driven upwards by the 
partial pressure of the enclosed gas, while this partial pressure 
was lacking in tube XIV. 

Theoretically, the meniscus should have slowly risen in the 
bell of tube XIV. owing to the pressure of the liquid ether 
upon the vapour in the bell. From the data given by 
Ramsay and Young (9) may be calculated the difference in 
temperature of the two menisci which is necessary to maintain 
them in equilibrium at a given difference of level. The 
difference of temperature is 0°-0008 C. for 1 cm. difference 
of level, or 0°-0064 C. for the difference of level of 8 cm. 
which existed in the tube. ‘The difference was probably a 
little greater than this, for very occasionally a bubble escaped 
from the mouth of the bell, showing that slow vaporization 
was in progress, Thus the tube acts as a delicate differential 
thermoscope whose delicacy increases as the mean temperature 
rises, since not only does the density of the ether decrease 
with rise of temperature but the slope of its vapour pressure 
becomes steeper. The difference of temperature persisted in 
spite of all precautions taken. About one-tenth of this 
temperature variation may be ascribed to the effect of the 
hydrostatic pressure of the aniline vapour upon its own 
tension; the remainder must be due to other causes. 

If the temperature of the tube was suddenly raised, the 
meniscus quickly travelled a certain distance up the bell and 
then remained practically stationary. Doubtless the vapour 


Critical Phenomena of Ether. 809 


in the bell, being supplied with heat less rapidly than that 
above, maintained equilibrium of temperature by partial con- 
densation. By suitabie manipulation it was possible near 
the critical temperature to have two menisci in the tube at 
different levels—one in the bell and one above the plunger. 
It was thus unnecessary to bring the meniscus down from 
above as originally contemplated. If the temperature was 
then raised a little above @,, sensibly the same phenomena 
occurred in both menisci as have been described in 
Section A(4). Hence there were two levels differing by 
about 6 cm. at which zones of transition appeared and per- 
sisted for several minutes until the opalescence became 
uniform. Nor did the zone in the bell show any perceptible 
tendency to rise. 


These phenomena favour the conclusion that the part played 
by hydrostatic pressure in the production of the transition 
zone is quite a subsidiary one. According to Gouy’s pre- 
diction there may be one level in the tube at which = is 
very great, but there can be only one such level. The exist- 
ence of temperature differences in the vapour jacket, it is 
true, makes this conclusion somewhat uncertain. If the 
variation of temperature with height is uniform, the only 
effect will be a tendency to annul the effect of hydrostatic 
pressure and to prevent the formation of the transition zone ; 
but if, owing to the separation of the tube into two sections 
by the plunger, each portion of the ether tends to assume a 
uniform temperature which is slightly higher for the lower 
portion than for the upper, the portions may be considered 
as representing short lengths of two isothermals in Andrews’s 
diagram separated by a very small temperature difference ; 
and in this case it is conceivable that each column may 


include the level at which et is a maximum for the given 


isothermal. It is probable, however, that, as in Villard’s 
experiment (v. page 794), this level of maximum value for 


is was not contained within the limits of the tube. The 
behaviour of the meniscus at its appearance during slow 
cooling, or its disappearance during very slow rise of tem- 
perature, seems to show that the mean specific density was 
so much less than the critical density that the level in 
question would be somewhat below the bottom of the tube. 


Phil. Mag. 8. 6. Vol. 20. No. 119. Nov. 1910. 3H 


810 Mr. F. B. Young on the 


The possibility of making the zone of transition form at 
varying heights (v. Section C. 4) in the tube also seems to 
point to the inadequacy of the explanation of Gouy. 


E. Influence of Difference of Temperature. 


The experiments to be described were practically repe- 
titions of Villard’s experiments with ethylene. By passing a 
current of electricity through the coil N (fig. 4) it was 
possible to raise the temperature of the upper half of the tube 
slightly above that of the lower half. In the experiments 
which were carried out with tube XIV. the lower limit of 
the coil was about 3 cm. above the top of the plunger. The 
variations of optical density in the ether were observed, when 
possibie, by means of a strip of squared paper placed behind 
the vapour jacket. The appearance of the lines when viewed 
through the tube maintained at a temperature slightly below 
6. is shown in figure 5A; the vertical lines are sharply 


Fig. 5. 


broken at the meniscus. If the temperature is raised rapidly 
to some point above @, the broken verticals become connected 
by a curve as shown in B; the curve gradually spreads 
upwards and downwards, and at the same time the upper and 
lower portions of the lines gradually open and close respec- 
tively until their appearance is uniform throughout. When 
the presence of opalescence made the observations difficult or 
impossible, the degree of uniformity of density of the ether 


Critical Phenomena of Ether. 811 


was judged by the degree of uniformity of the opalescence ; 
this method is justified in Section F. 

The temperature of the tube was raised a little above 0, 
and maintained for a few minutes. The plunger was 
vigorously raised and lowered several times in order to 
ensure complete uniformity. The ether was then uniformly 
opalescent throughout the tube. A current of about 
0-5 ampere was then passed through the wire coil. It was 
observed that the opalescence gradually faded away whilst 
the lines seen through the tube assumed the appearance of 
fic. 5 B ; the transition zone was just beneath the lower limit 
of the coil. After the heating current had been switched off, 
the appearance of the tube remained unaltered for about three 
minutes ; a faint opalescence then appeared in the transition 
zone which soon spread and filled the lower part of the tube. 
The upper part of the ether then grew opalescent, and finally 
the appearance of the tube became the same as it had been 
before the coil was heated. The phenomena scarcely differed 
from those described in Section C (4). 

The transition zone was reproduced as before. The heating 
current having been switched off, the temperature was slowly 
reduced. It was observed that the transition zone became 
opalescent, the limits of the opalescence being sharply defined ; 
the opalescence grew denser until at length a meniscus 
appeared within it. The opalescence then faded away. The 
phenomena of Section C (5) were thus reproduced. 

An attempt was then made to compare the relative per- 
sistencies of the natural and the artificial transition zones. 
The temperature was suddenly raised from about 193°4 C. to 
193°-73 C. in order to produce the phenomena of Section C (4). 
The time was then taken which elapsed before the zone of 
transition had lost all definition and the opalescence had 
spread to the top of the tube; this proved to be about 
10 minutes. After a short interval during which the tem- 
perature remained at 193°°73 C., the heating current was 
switched on for two minutes, and the time was again taken 
which had elapsed before the appearance of the tube was 
sensibly the same as at the end of the former test : the period 
was about 8 minutes. 

The tests with the heating coil show it to be highly pro- 
bable that differences of density above @ which persist for as 
much as 8 or 10 minutes may be ascribed to residual tem- 
perature differences produced by vaporization. 

Mention may be made of two effects observed when the 
heating coil was used below the critical temperature. 

It was possible, by passing # stronger current through the 

3 H 2 


812 Mr. F. B. Young on the 


coil, to cause the meniscus to disappear when the temperature 
of the vapour jacket was as much as 8° below @,, so that the 
lower portion of the ether was still presumably in the liquid 
state. Owing to the great difference in density the transition 
zone was then extremely well marked. Its depth was so slight 
that it had almost the appearance of a convex meniscus. 
Closer examination, however, led to the conclusion that the 
appearance of convexity was due to the refractive effect upon 
horizontal lines placed behind (fig. 5), the effect being pro- 
duced by the presence in a cylindrical tube of a medium 
gradually increasing in density downwards. The apparent 
slight convexity of the meniscus mentioned in Section C (4) 
may have been due to this cause. The matter is here men- 
tioned because a convexity of the meniseus in the neighbour- 
hood of the critical temperature has previously been 
recorded (10) ; it is now suggested that in such cases the 
temperature @, had already been slightly exceeded and that 
the so-called meniscus was in reality a very narrow transition 
zone, 

If the bell was filled with vapour at a temperature con- 
siderably below @, and the heating current was then switched 
on, the liquid was observed to rise and fill the bell (the upper 
meniscus was for this purpose embraced by the coil). If, 
however, the temperature was not far below 6, the meniscus, 
after rising a short distance, became nebulous and faded 
away asitrose. A similar effect is mentioned by Amagat (11), 
who found it occur when carbon dioxide slightly below its 
critical temperature was subjected to slow compression. It 
was probably due to the evolution of heat by the vapour as it 
condensed under compression. By diminishing the heating- 
current and consequently the rapidity with which the 
meniscus rose, it was possible to make the meniscus rise 
higher before disappearing, the latent heat of vaporization 
having more time in which to dissipate. 


FE. Opalescent Phenomena. 


(1) Occurrence. — The opalescent effects described in 
Section E afford some information concerning the conditions 
which favour the production of opalescence. On reducing 
the temperature to @, after the production of the artificial 
transition zone, a dense opalescence appeared in that zone 
only, no visible phenomena occurring above or below as the 
temperature fell. Although the temperature of the ether 
varied through the tube, each layer of ether passed through 
8, in turn ; since marked opalescence was exhibited only by 
that part of the ether in which the meniscus appeared, and 


Critical Phenomena of Ether. 813 


which, therefore, presumably possessed the critical density, 
it seems that opalescence can occur in ether only if the 
density approximates to the critical density. Again, since 
conversely in every case in which the intensity of the opales- 
cence was not uniform the meniscus appeared in the part 
in which the opalescence was most marked, it may be inferred 
that the critical density is the density most favourable to the 
production of opalescence. 

Travers and Usher (7) observed that if the mean density 
of filling differs from the critical density, so that as @, is 
approached the meniscus is either rising or falling in the tube, 
then the disappearing phase becomes markedly more opales- 
cent than the increasing phase. They suggest an explanation 
based upon Donnan’s theory concerning the surface-tension 
of small drops, and connect the localization of the opalescence 
with the motion of the meniscus. This phenomenon was 
strikingly apparent in tubes XIII. and XIV. when they were 
raised simultaneously to the critical temperature; the liquid 
phase in XIV. became intensely opalescent before the vapour 
showed more than faint indications, whilst in tube XIII. 
the conditions were reversed. The phenomenon, however, 
recelves a simple explanation from the dependence of the 
intensity of the opalescence upon the density of the substance. 
Unless the mean specific volume is equal to the critical 
volume, the meniscus can only disappear within the limits of 
the tube if, at the time of disappearance, the two phases 
differ in density, and hence at @, only one of the phases can 
approximate to the critical volume. If the mean specific 
volume is less than V,, this phase will be the phase of lesser 
density, i.e. the vapour phase, which is also the diminishing 
phase ; it is this phase whose density will be more favourable 
to the existence of opalescence. The reverse is true if the 
mean specific volume is greater than the critical volume. 
This explanation is in effect identical with the conclusion of 
Sidney Young (12) that the position of maximum opalescence 
depends upon the mean specific volume, since of course the 
actual specific volume at any given height in the tube and 
the mean specific volume are interdependent. 

That the presence of the meniscus is unnecessary for the 
production of the phenomenon is shown by the effects 
described in Section C(4) where the local opalescence 
appeared after the fading of the meniscus ; under the con- 
ditions of Section C (4) the same striking difference was 
apparent between tubes XIII. and XIV. The same indeed 
was true when the transition zone pruduced above 6, by 
means of the heating-coil was allowed to disappear. The 


814 Mr. F. B. Young on the 


effects of traces of permanent gas upon the intensity of the 
opalescence are mentioned in Section F'; it seems probable 
that these effects are indirect and consequent upon the 
influence of the impurity upon the density of the ether. 

The variation of the intensity of the opalescence with tem- 
perature was examined. It was observed that the intensity, 
which was greatest at @,, decreased as the temperature 
steadily rose above or fell below that temperature but 
decreased at a diminishing rate, so that the opalescence dis- 
appeared insensibly. It was, therefore, difficult to determine 
the limiting temperatures, particularly as the opalescence 
was viewed through six layers of glass. The lower limit, 
however, appeared to be about 0°5-0°°6 below 6,, whilst there 
was a suspicion of opalescence at the highest temperature 
reached, 2. e. about 3° above @,. 


The Nature of Opalescence. 


Various explanations of the nature of the opalescence have 
been suggested. 

Some investigators have considered it to consist merely of 
an emulsion of one phase in the other (13, 14). Just below 
_ the critical temperature the liquid and vapour differ so little 
in density that either phase may exist dispersed throughout 
the other in the form of fine particles. Donnan (15) supports 
this view, but suggests certain conditions of surface tension 
which might account for the stability of the opalescent 
particles. 

Kiister (16) suggests that the opalescence is due to the 
variation in the temperatures of the individual molecules 
according to the kinetic theory. Let the mean temperature 
of the substance be just above the critical temperature ; then, 
according to the theory of probabilities, many of the mole- 
cules will be moving with a velocity much lower (or higher) 
than corresponds to the mean temperature. When a sufficient 
number of slow-moving molecules occur together for an 
instant they constitute a minute drop of liquid. Any 
individual drop will only have a momentary existence 
since by hypothesis the drop has a lower temperature than 
its surroundings, but new drops will be created incessantly 
throughout the substance. In this way the opalescence above 
the critical temperature is accounted for ; the opalescence in 
the liquid and vapour phases just below the critical tempera- 
ture may readily be explained in a similar manner. 

The theories stated have some interest in connexion with 
the present inquiry inasmuch as they implicitly assign 


; 

| 
q 
a 


Critical Phenomena of Ether. 815 


different values to the critical temperature. Whilst neither 
of the theories makes any assumption either in favour of or 
contrary to the liquidogenic hypothesis, the theories of 
Altschul and Donnan imply that the Cagniard-Latour tem- 
perature is below either the temperature of complete mutual 
solubility of the liquidogenic hypothesis or that of uniformity 
of state of the simple classical theory ; Kiister’s theory, how- 
ever, would imply that the Cagniard-Latour temperature 
coincides with one or the other of these. The fact that 
ebullition may proceed concurrently with the existence of 
opalescence of considerable intensity (v. Section C(16)) is 
scarcely in accord with Altschul’s simple explanation, since 
it implies that an appreciable difference of density exists 
between the phases. An observation of the condensation 
phenomena also leads to the conclusion that there is a differ- 
ence in kind between the stable opalescent cloud observed 
above or below @& and the cloud of condensation produced at 
the moment of separation of the meniscus. If the tube, first 
raised above @,, is steadily and fairly slowly cooled down, the 
opalescence increases in intensity as @, is approached but 
retains a “dry” nebulous appearance. At the moment of 
appearance of the meniscus, however, the condensation cloud 
appears to be superposed upon the opalescent cioud, some- 
times flashing through the tube, sometimes spreading rapidly 
from the bottom upwards. This is particularly evident when 
the opalescence is localized, by the various means described, 
to a shallow zone. The “dry” fog instantaneously changes 
to a “ wet” fog in a manner difficult to describe. When the 
condensation cloud has settled, the opalescent cloud is still 
visible enclosing the meniscus. The opalescent particles 
probably supply the condensation nuclei, but seem themselves 
to be something other than mere liquid drops or vapour 
bubbles. This phenomenon may perhaps be explained in 
accordance with Donnan’s theory, if itis supposed that under 
suitable conditions of cooling the increase in the bulk of the 
opalescent material may tend to proceed rather by the increase 
in the size of the existing particles than by the formation of 
new aggregates, and that at length a certain proportion grow 
beyond the limits of size for which the peculiar conditions of 
surface tension which secure stability are true. 

The consideration of the intensity of opalescence as a 
function of the temperature may, however, afford some 
criterion in deciding between the theories stated :— 

Let a tube of capacity V contain 1 g. of the substance at 
a temperature below @,; if the volume of liquid is v whilst 
u and uw are the specific volumes of liquid and vapour 


816 Mr. I’. B. Young on the 


respectively, then the proportion by volume of the liquid 
present is given by 
oy a 


VO V(uy—u)’ 
Figure 6 shows the graph of = with temperature when V 


is greater than V,. The graph is plotted from the experi- 
mental values of wu and uw; given by Ramsay and Young (20), 


Fig. 6. 


160 16S Igo igs °C 


Tem p > 


V being taken as 4°5, but its shape is characteristic of 
such a graph derived from a saturation curve of the form 
given in Andrews’s diagram, 2.e. a simple curve of negative 
curvature. It shows that the volume of the liquid diminishes 
with increasing rapidity, the temperature of disappearance 
being sharply marked. A similar graph would be obtained 


by plotting = with temperature in the case where 


V<V,. According to Altschul’s simple explanation, either 
the bulk of opalescent material present should disappear in a 
similar manner with increasing rapidity and a well-marked 
point of disappearance, or the accepted form of the saturation 
curve requires modification and that curve must be given a 
peaked form. The assumptions of Donnan concerning 
surface-tension do not seem greatly to affect the nature of 
the graph. The optical intensity of the opalescence will, it 
is true, doubtless depend not only upon the bulk of the 


Critical Phenomena of Ether. 817 


opalescent material but also upon the difference of optical 
density between that material and the surrounding medium. 
Since the optical densities of the liquid and vapour, however, 
tend to equalize with increasing rapidity as 0, is approached, 
the sharpness of the point of disappearance would be still 
further accentuated. 

It was observed that the intensity of the opalescence above 
the Cagniard-Latour temperature decreased rapidly at first 
with rise of temperature, but with diminishing rapidity as 
the temperature became higher. To obtain some experi- 
mental record a rough graph showing the intensity of the 
opalescence at various temperatures was made in the following 
manner. A number of tubes of similar dimensions were 
filled with water containing a proportion of milk varying 
from 0:02 to 10 per cent. If one of these was held before 
the plate-glass observation window in a suitable position and 
under suitable conditions of illumination, an image could be 
obtained, close to the experimental tube, which resembled 
very closely a tube of opalescent ether. The series of tubes 
formed a rough scale of opalescence, with which the opalescent 
ether was compared at various temperatures. In fig. 7 the 

Fig. 7. 


c 


40 Go 80 106 


— Intensity of Opalescence—> 
20 


1G3-6 nes. 194-0 194-2 194-4 1G4-6 1g&9 °C) 
=== Temp. _ 


ordinates represent the intensity of the opalescence in terms 
of the proportion of milk contained in the tube whose image 
matched it most nearly (10 per cent. milk =100). The curve 
shows sufficiently well the manner in which the opalescence 
varied with the temperature, and the form of the curve is 
altogether different in character from that of fig. 6. 

The curve of opalescence is rather of the character which 
might be predicted from Kiister’s theory. According to that 
theory, there would probably be some connexion between 
the intensity of the opalescence and the distribution of velo- 
cities of translation according to Maxwell’s theorem. The 


818 Mr. F B. Young on the 


proportion of molecules having less than a given velocity hk, 
which is itself less than the mean velocity of translation K, 
diminishes as & is taken further from K, but the rate of 
diminution is less rapid as (K—k) becomes greater. This 
matter will, however, probably be further investigated. 


G. Lffects of Impurity. 


Dwelshauvers-Dery (17) maintained a tube of carbon 
dioxide at various temperatures above @, and then allowed 
the tube to cool. A zone of emulsion appeared in the neigh- 
bourhood of the point at which the meniscus had disappeared, 
and as the temperature to which the tube was raised became 
higher, the zone of emulsion which was formed on cooling 
became broader. The emulsion, however, was not uniform 
throughout the tube even when the temperature had been 
raised 20° above @,. He inferred from his observations of 
carbon dioxide that the two states of the substance persisted 
even to the highest temperature reached. The zones of 
opalescence were regarded by him as zones in which mutual 
diffusion of the gasons and liquidons had occurred, so that 
when the temperature was lowered a cloud was formed by 
the separation of the two kinds of molecules. 

Andrews (18), however, has obtained similar opalescent 
zones in working with mixtures of carbon dioxide and small 
proportions of nitrogen, though under rather different experi- 
mental conditions. The similarity of these results suggested 
that the phenomena of opalescence might be closely associated 
with, if not entirely dependent upon, the presence of impurity. 
It was partly with the object of testing this that special 
efforts were made to free the ether from impurity. 

It was found, however, that the formation of opalescence 
is retarded rather than assisted by the presence of impurity. 
As the ether was obtained purer it more readily showed signs 
of general opalescence when its temperature was raised. 
When tubes VII. and XIV. were together raised in tempera- 
ture slightly above @,, some time after the opalescence in 
XIV. had become general, tube VII. showed only a slight 
opalescence throughout the tube, whilst a denser cloud 
appeared in the neighbourhood of the point at which the 
meniscus had faded. If the ether was stirred, however, both 
tubes immediately appeared equally opalescent. The influence 
of traces of impurity upon the opalescent phenomena is pro- 
bably indirect. If a trace of gas is present in the upper 
part only of the tube, it produces, by its partial pressure, a 
difference of density of the ether which may be so great that 


Critical Phenomena of Ether. 819 


only the layers of ether at the junction of the two masses is 
of the density most favourable to the production of opal- 
escence. The rate at which the opalescence spreads is 
dependent upon the rate of diffusion of the gaseous impurity, 
which is very slow. Even the slight impurity contained in 
tube XIII. seemed sufficient to delay visibly the establish- 
ment of equilibrium. If tubes XIII. and XIV. were raised 
simultaneously slightly above 6, both tubes quickly showed 
opalescence in one portion (in tube XIII. this was above the 
transition zone), but the opalescent column in XIII. remained 
well defined for a few minutes after that in XIV. had become 
diffuse. It appears then that the readiness with which the 
opalescence becomes uniform throughout the substance, when 
the tube is heated slightly above @,, may be taken as a 
criterion of the freedom of the substance from gaseous 
impurity. 

For the purpose of exaggerating the effects of impurity 
a tube XVII. was filled with ether contaminated with 5 per 
cent. of alcohol. The air was expelled from the tube, pre- 
paratory to sealing, by boiling off the excess of ether over 
mercury. The ether was heated, however, near the surface, 
so that the lower layers were unaffected and retained the 
dissolved air. The critical temperature 0, for this tube was 
approximately 196°°2 C. It was maintained at the highest 
temperature for which the thermostat was constructed (about 
196°:7 C.) for 80 mts. At the end of that time a very well- 
marked transition zone was still made evident by the squared 
paper placed behind ; it extended through a height of about 
10 mm. only and was slightly opalescent. The temperature 
was then very slowly reduced, the reduction of 0°°5 occupying 
about 30 mts.; at the same time the transition zone was 
watched carefully through a telescope containing a scale eye- 
piece. It was observed that the transition zone contracted 
slightly in height, whilst the relative displacement of the 
upper and lower parts of the vertical lines became more 
marked. The opalescence in the zone became gradually 
denser and more sharply defined. At the same time it be- 
came shallower: at 196°4 C. it was a dense cylinder about 
2°5 mm. in height, and at 196°-2 C. its height was scarcely 
1mm. The opalescence appeared to have contracted into 
the upper part of the transition zone, for the curvature of the 
vertical lines was still visible underneath. At 196°2 C. the 
meniscus appeared in the opalescent zone. The transition 
zone had persisted for an hour, and at the end of that time 
was still clearly marked. The concentration of the opal- 
escent material was probably only apparent, and the zone of 


820 Mr. F. B. Young on the 


opalescence became narrower simply owing to the increase in 
the slope of density through the transition zone, produced by 
the fall in temperature ; when uniformity of density was 
produced by stirring, the opalescence likewise became uniform 
and showed no tendency to become localized. The opalescent 
effects produced without stirring were very much like those 
produced in tube XIV. with the aid of the heating-coil, but 
were far more lasting. 

It is necessary to observe caution in applying the results 
obtained with one substance to the explanation of phenomena 
observed in another. So far, however, as this may be done, 
it seems probable that the phenomena observed in tubes of 
carbon dioxide by Dwelshauvers-Dery might readily be pro- 
duced if the experimental substance contained strong traces 
of some much more volatile impurity. It is unnecessary to 
regard the opalescent zone as a zone of separation of impurity 
from the ether, or the opalescence as consisting of the 
impurity in suspension. The opalescence is conditioned by 
the diffusion of the impurity only in so far as such diffusion 
produces a graduation of density in the substance, and hence 
the intensity of the opalescence or the bulk of opalescent 
material may be altogether out of proportion to the quantity 
of impurity present. 


H. The Liquidogenic Theores. 


Though it cannot be claimed that the experiments recorded 
by any means settle the liquidogenic controversy, yet they 
have some evidential value. 

Travers and Usher (7) have criticised experimentally the 
results of Battelli by means of which he showed that, in the 
case of ether, 0, decreases progressively as the mean density 
of filling increases. It is to-be observed that the mean den- 
sities of filling chosen by Travers and Usher were contained 
within much narrower limits (0°244—0°281) than those taken 
by Battelli (0°2409-0°3043). Though the former investi- 
gators do not explicitly state the fact, one’s own observations 
lead to the conclusion that if the tubes approaching the 
higher limit taken by Battelli were filled with ether sensibly 
pure and free from permanent gas, it would be impossible to 
make the meniscus disappear within the limits of the tube 
without raising the temperature of the thermostat so rapidly, 
in order to exaggerate the Villard effect, as to make accurate 
observations impossible. In tube XIII. it was found that 
with sufficiently slow rise of temperature the meniscus rose 
to within *5 cm. of the top of the tube, though the mean 


Critical Phenomena of Ether. 821 


density of filling was only 0°285 (density of ether at 0° C. 
being taken as 0°7362). If, however, sufficient traces of per- 
manent gas or other impurities were present the task would 

_ be simple—it is quite probable that in tube XIII. the reluc- 
tance of the meniscus to ascend to the top was due to the 
trace of permanent gas known from the tests of Section A to 
be present—but in this case the results would no longer 
necessarily apply to the pure substance. 

The slight difference in 0, for the tubes XIII. and XIV. 
is not to be taken as iending support to Battelli’s results. 
The values given in Table 1. are for the appearance of the 
meniscus with fall of temperature. If the ordinates, or lines 
of constant volume, are traced on Andrews’s diagram for 
various values of v in the neighbourhood of V,, it becomes 
evident that since the separation into two phases can only 
occur when the given ordinate reaches the saturation curve, 
the meniscus should appear at temperatures varying with the 
specific volume and diminishing as the specific volume differs 
more from V,. Since in tubes XIII. and XIV. the meniscus 
appeared invariably at the upper and lower extremities of the 
respective tubes, V, evidently lay between the two specific 
volumes,and if it lay nearer to the specific volume of tube XIV. 
the slight difference of temperature might be predicted from 
Andrews’s theory. It was difficult to determine the exact 
temperature of disappearance on raising the temperature, 
partly because of the gradual nature of the change, partly 
because of the opalescent effects which rendered observation 
difficult, but in general the temperature of disappearance in 
both tubes seemed slightly higher than that of appearance, 
and was certainly not lower for tube XIII. than for 
tube XIV. 

Other observers, like Battelli, have observed remarkable 
differences of density at @, or slightly above. Since the differ- 
ence in the mean densities of tubes XIII. and XLV. is about 
11 per cent. and the menisci disappeared practically at the 
top and kottom of the respective tubes (given a sufficiently 
gradual rise in temperature), it follows that at the tempera- 
ture of disappearance the difference of density could not 
have exceeded that amount. 

Mathias (19) shows, however, in his analysis of the liquido- 
genic theories that it is quite possible to frame a theory 
which will coincide with the classical theory of Andrews in 
its predictions concerning the final states of equilibrium, 
whilst it will also account for the transitory differences of 
density which have been held to show the insufliciency of 
that theory. It is the theory according to which the ratios 


q 


$22 Mr. F. B. Young on the 


h, and A of gasons to liquidons in the vapour and liquid phases 
respectively are in the final state functions of the temperature 
only, the functions being different for the respective phases. 
If the law of Avogadro is assumed to hold for both kinds of 
molecules, the temperature of uniform density is that at 
which the ratios become identical. As the temperature is 
raised the establishment of equilibrium involves a continuous 
reduction in the value of h, and a continuous increase in the 
value of 4; the adjustment is produced by a combined process 
of diffusion and of transformation of one kind of molecule 
into the other. A lag in this adjustment will sufficiently 
account for the prolonged differences of density which have 
been observed. 

Since this theory differs from the classical theory in its 
predictions concerning observable phenomena only in respect 
to the time required for the establishment of equilibrium, it 
is only possible to decide between them by observation of the 
time-factor. Though the fact that the transition zone may 
be reproduced by heating the upper portion decisively proves 
that diffusion must play a minor part in the establishment of 
uniformity, it cannot be regarded as crucial evidence agains 
the validity of the liquidogenic theory: the objection may 
reasonably be raised that the changes of temperature and 
pressure which are produced tend to produce dissociation in 
the heated portion and possibly association in the lower 
portion. It is, however, doubtful whether the time-factor 
must be necessarily greater on the assuinption of the liquido- 
genic hypothesis than on the assumption of Andrews’s theory. 
According to the latter, if we accept Villard’s explanation of 
the transition zone as of predominant importance, the denser 
substance must expand and the less dense substance be com- 
pressed against the external pressure and the intramolecular 
forces combined: this involves virtually the transmission of 
energy from the upper to the lower part of the substance, 
the transmission probably occurring mainly through the walls 
of the tube. In addition to this, sufficient heat must pass to 
bring the substance to the temperature of the thermostat. 
The liquidogenic theory seems to differ from Andrews’s theory 
only in substituting the force of chemical affinity for the 
cohesive force, and the time-factor involved will only differ 
greatly on the assumption that the energy absorbed in the . 
disintegration of the liquidons is much greater than that 
absorbed in separating the molecules against the cohesive 
force. : 

In Table II. are given readings for plotting by means of 
Van der Waals’s equation the isobar which passes at the 


Critical Phenomena of Ether. $23 


TABLE II. 
v (normal yols.). | rs t—194°55. SDs b 

Cc. 
0:0170320 467°504 —0:046 5DD 
‘0172320 | 467-521 — -029 4-44 
0174320 467°535 — -015 3°33 
‘0176320 | 467°542 — -008 2-22 
‘0178320 467°547 — -003 111 | 
-0180320 (Ve) 467550 000 0-00 
‘0182320 | 467°552 + -002 ioe 
0184320 467°556 + -006 2-22 
0186320 467°563 + 013 _ 3:33 
‘0188320 | 467-574 + 7622") 4-44 | 

J 


critical volume through the temperature @.+0° 15. The con- 
stants a and b, as well as the theoretical critical volume V,, 
are derived from the critical data given by Young (20) 
(@-=194'4; P,=35°61 atmospheres), and are a=0:0347314, 
b=0°00601066, V,=0:0180320. The temperatures (¢) were 
then calculated for the volumes given in the table from the 
formula 


t=(p+ >)(v—b)/R, 


R(273+194:55) a 
(V.—b) ey yee 


From the table it appears that a difference of temperature 
of 0°:014 C. between the two portions of the substance may 
produce a difference of density of about 4°5 per cent. Owing 
to the great pressure under which the changes of density 
occur and the small temperature slope through the glass walls 
of the tube, it may be expected that the establishment of 
equilibrium will be a slow process. 

Tube XIV. was raised to a temperature of about 0°13 C. 
above @,; the time which elapsed before the opalesvence 
became absolutely uniform in appearance was 14 minutes. 
This does not seem an unduly long period for the establish- 
ment of equilibrium and seems to render a liquidogenic 
hypothesis superfluous. 

It must be acknowledged that the slight differences of 
temperature occurring in the thermostat would tend to acce- - 
lerate the establishment of equilibrium, both by increasing 
the rate of transmission of heat through the walls of the tube 
and by producing convection currents, though the latter were 
not perceptible. Some quantitative knowledge of the relation 


where p = 


824 Mr. F. B. Young on the 


between the intensity of the opalescence and the density of 
the substance is also desirable, but owing to the difficulties 
of purification of the ether and of exact measurements at the 
critical temperature, this of itself would involve lengthy 
research. 


GENERAL CoNCLUSIONS. 


The experiments described have perhaps served rather to 
demonstrate the difficulty of realising theoretical conditions, 
than to lead to a decisive conclusion concerning the nature 
of the Cagniard-Latour phenomena. The results of the 
investigation, however, clearly favour the retention of the 
classical theory of Andrews. 

It has been found that by the introduction of a temperature 
difference, the Cagniard-Latour phenomena may be repro- 
duced with similar opalescent effects and approximately the 
same persistency. 

The last traces of gaseous impurity have proved most 
difficult to eliminate, and extremely slight traces of such 
impurity have visibly accentuated the phenomena or delayed 
the establishment of equilibrium. 

Since the differences of temperature involved are so slight, 
the period required for the establishment of equilibrium does 
not seem unduly long. 

The intensity of the opalescence depends greatly upon the 
density of the substance, so that any cause which produces 
a small difference in density will produce visible qualitative 
effects which are great in proportion to that cause. 

The investigation of the nature of the opalescence, so far 
as it has been carried, tends to favour the kinetic explanation 
of Kiister rather than that of Altschul or Donnan, and thus 
far decides the Cagniard-Latour temperature to be the true 
critical temperature of vaporization. 

It may be noted that the more recent investigations have 
in general tended to show that the more violent discrepancies 
from the predictions of Andrews’s theory concerning the 
critical phenomena have been due to causes which are not 
out of accordance with Andrews’s theory. Owing to the 
peculiar properties of the substance at the critical tempera- 
ture, the approach to conditions which permit of theoretical 


results must of necessity be asymptotic, and therefore to 


formulate an hypothesis for the purpose of explaining such 
discrepancies as may now be considered to remain seems un- 
necessary, except perhaps in the case of those liquids whose 
surface-tensions point to association of their molecules 


Critical Phenomena of Ether. 825 


APPENDIX. 

A Vapour Thermostat for work on Critical Phenomena. 

In connexion with the vapour jacket shown in fig. 3 an 
arrangement was employed for maintaining the vapour at 
constant pressure, so that constant temperature was secured 
(fig. 8). 

The vapour jacket communicates, through a large ballon 
B, with the closed mercury manometer C. The manometer 


Fig. 8. 


m 


——_e_e— ew @ = 


-—— te ee ee ee ee ee ee eee ee ee 


—e_ wee Oe eww eK 


is fitted with a Topler siphon-tube, so that the vacuum may 
be tested and, if necessary, renewed. In the short arm of 
the manometer is a float E, which makes electrical contact 
with a platinum-tipped needle F. The float consists of a light 


Phil. Mag. 8. 6. Vol. 20. No. 119. Nov. 1910. aI 


826 Mr. F. B. Young on the 


copper disk to the top of which is cemented a piece of thin- 
walled glass tubing, constricted near the bottom. On the 
top of the copper float is soldered a piece of platinum-foil ; 
the under surface is amalgamated. When contact is made, 
a relay G is actuated and a current of about 1 ampere is sent 
through a 5-ohm coil of fine german-silver wire H contained 
in the ballon B and lightly wrapped in cotton-wool. The 
heat developed causes the air to expand until the consequent 
rise of pressure breaks the contact at H. If the temperature 
of the ballon is already rather higher than that of the atmo- 
sphere, the pressure again falls owing to radiation of heat, 
until contact is again made. The cotton wool has the effect 
of making the expansion less sudden, so that the mercury in 
the gauge may follow the changing pressure more closely. 
Since the height of the mercury in C can be regulated by 
means of the reservoir D, any desired pressure can be main- 
tained in the apparatus. 

A slight tendency to stick on the part of the contact E is 
corrected by means of the electromagnetic trembler K, which 
is placed as a shunt in the heating circuit. Its base-board is 
attached to the needle F by a wire whose tension is suitably 
adjusted. The tendency is reduced toa minimum by deli- 
cately poising the relay key and reducing the relay current 
to a minimum. 

Tests have shown that the contact can be relied upon to 
make and break within a range of 0°035 mm. movement of the 
float, the maximum oscillation of pressure due to this cause 
being therefore 0°07 mm. of mercury ; in general the oscil- 
_lation is less than this. The troublesome effects due to the 
surface tension of the mercury are practically eliminated by 
making the manometer tubes rather wide (about 2 cm.), and 
by introducing a thin layer of ‘ Fleuss’ oil over each mercury 
surface. It is also advisable to avoid depressing the float 
into the mercury during the process of adjustinent. 

The manometer is made independent of temperature varia- 
tions by a suitable adjustment of its dimensions. Let P be 
the indicated pressure of mercury, S the area of cross-section 
of the longer limb, V the total volume of mercury, c the 
coefficient of cubical expansion of mercury, and a the coeffi- 
cient of linear expansion of glass. Then, assuming the float 
to remain unmoved relatively to the glass tube, the change of 
pressure dp due to a change of temperature dt is given very 
approximately by 
V(c—3a) . dt 

S 


api +a.P.dt—c.P.dt. 


Critical Phenomena of Ether. 827 
The pressure is unaffected if 


_ PS(c—a) . 
i. eae Ase tus Gey AS eyes wahee aD 

The movement of the float produced by the expansion of 
the needle is negligible for moderate lengths of the latter, 
since it depends upon the difference between the expansion 
of glass and that of steel. 

Hquation (I.) implies that if adjustment has been obtained 
for a given pressure P, in order to obtain adjustment for a 
pressure P +p, the increase v in the volume of mercury must 
be pts 8) é 

c—3a 
of constant and equal cross-section, the point of the needle 
pa 


¢—3a 


If the two menisci move in cylindrical tubes 


must be raised through a distance At -p) or 


(=p x 0:0554). 

In practice the needle was soldered into a brass cap at the 
height corresponding to P=965 mm., which is the pressure 
of aniline vapour in the neighbourhood of the critical tem- 
perature of ether. 

L is a narrow graduated tube which forms a subsidiary 
reservoir for fine adjustments of the mercury. If D is dis- 
connected by closing slip M, small measured amounts of 
mercury can be introduced into the gauge or withdrawn 
from it. In this way it is possible to raise or lower the 
temperature of the thermostat by successive steps of 0°01 or 
less, the pressure being adjusted by the heating-coil. 

For considerable changes of pressure the reservoir D must 
be used, and air must be pumped into or withdrawn from the 
ballon through the three-way tap N. The heating-coil will 
satisfactorily maintain constant pressure only if the tempera- 
ture of the air in the ballon as indicated by thermometer P 
is within certain limits (5° to 15° above the atmospheric 
temperature). 

The irregularity of boiling adds considerably to the oscil- 
lation of pressure. In order to secure the best results it is 
necessary to employ a gas supply of steady pressure and to 
introduce broken porous ware into the aniline in the usual 
manner. 

In a test lasting about three hours, during which the 
pressure was observed for a few minutes at frequent intervals 
by means of a water gauge, the extreme variation of pressure 
observed was ‘29 mm. of mercury: this corresponds to a 
variation in temperature at 193-4° C. of 0°12 ©. If one 
reading is omitted, the variation was 0°19 mm. or 0°098 C. 

a1.2 


825 Dr. J. W. Nicholson on the 


The oscillation of the pressure did not exceed 0°12 mm.; 
the period of oscillation (7. e. from contact to contact) varied 
from 6 sec. downwards. 


In conclusion I desire to express my thanks for the 
facilities afforded me by the Physical and Chemical Depart- 
ments of the University of Bristol, at which the research 
was conducted. I am greatly indebted to Professor A. P. 
Chattock and Dr. James W. McBain for constant advice, 
and to the Jatter for much personal assistance. 


REFERENCES. 


. Gouy, C. #., t. exv. p. 720 (1892). 

. Villard, C. #., t. cxxi. p. 115 (1895). 

Communications from Physical Laboratory at University of Leyden, 

No. 68, 1901; and Supplement 10, 1904. 

. Cailletet & Collardeau, C. #., t. eviii. p. 1280 (1889). 

. Hagenbach, Ann. d. Physik, (4) B. v. 8. 276 (1901). 

. Eversheim, Ann. d. Physik, (4) B. xiii. S. 492 (1904). 

. Travers & Usher, Proc. Roy. Soc., A, vol. lxxvill. p. 247 (1906). 

. Wade & Finnemore, Journal of Chem. Soc., Nov. 1909, p. 1842. 

. Ramsay & Young, Phil. Trans., A, vol. elxxviil. p. 57 (1887). 

. Wolf, Mathias’s “ Point Critique,” p. 124 (Ann. de Ch. et Phys. (3) 

t. xlix. p. 270, 1857). 

11, Amagat, C. &.,, t. exiv. p. 1093 (1892). 

12. S. Young, Proc. Roy. Soc., A, vol. lxxviii. p. 262 (1906). 

13, Altschul, Zect. Phys. Chem., B. xi. S. 578 (1893). 

14. Ramsay, Zet. Phys. Chem., B. xiv. 8. 486 (1894). 

15. Donnan, Brit. Assoc., Section B, 1904, p. 504. 

16. Kiister, Lehrbuch der physitkalische Chemie, p. 1907. 

17, Dwelshauvers-Dery, Mathias’s ‘‘ Point Critique des corps purs,” 
p. 280 (Bull. de [ Acad. roy. de Belgique, 3e série, t. xxxi. p. 277, 
1896). 

18, Andrews, Phil. Trans., A, vol clxxviii. p. 45 (1887). 

19, Mathias’s * Point Critique des corps purs,” p. 218, Theorie ii. 

20. S. Young, Phil. Mag., vol. 1. p. 291 (1900). 


oo bo 


— 
COMNIMA UF 


XC. The Accelerated Motion of a Dielectric Sphere. 
By J. W. Nicuousoy, M.A., D.Sc.* 


[ a previous paper, a brief account was given of the 
motion of a conducting sphere whose mass is purely 

electrical, under the action of either a small uniform tield of 

electric force or a small mechanical force. The solution was 

deduced as a limiting case from a more general problem 

treated by G. W. Walker {, and it was shown that there are 

difficulties in the results of regarding any conductor as 
* Communicated by the Author. 


+ Roy. Soc. Proc. A. vol. Ixxvii. p. 260 e¢ seg.; Phil. Trans. A. 
1910, p. 145 ef seq. 


Accelerated Motion of a Dielectric Sphere. 829 


perfect when its motion is accelerated. The perfect con- 
ductor of the usual theory leads to disturbing infinities when 
it has no Newtonian mass. The indications that the mass of 
a single electron can have a Newtonian element are not very 
securely established ; and although certain experiments can 
be interpreted in accordance with this view, there is always 
a possibility of other interpretations which do not involve it. 
For example, it is possible that the particles in Kaufmann’s 
experiments are electrons not free, but attached to matter. 
A comprehensive examination of the conditions of motion of 
a small body without a Newtonian mass is therefore desirable, 
and this was made in the case of a conductor under the action 
of a small force in the previous paper. Apart from indica- 
tions there obtained, it seems unlikely on general grounds 
that an electron can be endowed with properties analogous 
to those of a conductor, for there is a difficulty of attaching 
a physical meaning to such properties in a single electron. 
Moreover, the rapidity of damping of the oscillations set up 
when the motion of the conductor is changed, supplies a 
strong adverse argument. 

Some interest therefore attaches to the corresponding 
problem of a small sphere, with a surface charge, whose 
interior has the properties solely of a dielectric, with no 
conducting element. 

In the present paper, the motion of such a sphere, devoid 
of Newtonian mass, is investigated, and it is shown to present 
none of the difficulties noticed in the case of the conductor. 
A small field of force can produce a finite acceleration, and 
will give the effect of a constant acceleration after a short 
time, if the dielectric coefficient be not too great. If this 
coefficient is great, the oscillations initially set up are very 
permanent, and the constant acceleration is not established 
by a uniform field within the time during which the equa- 
tions are good approximations to the motion. The problem 
in this case bears some resemblance to that of the perfect 
conductor, for the disturbance inside the sphere tends to 
zero as the dielectric constant increases. But the problems 
do not become identical, for in the case of the conductor, the 
charge is allowed freedom of movement on the surface, and 
in fact does redistribute itself in the manner previously 
calculated. In the dielectric sphere, it remains uniformly 
distributed, and the problem thus corresponds to those of 
accelerated motion usually treated by the quasi-stationary 
principle, in whose application any redistribution is ignored. 

The main outlines of the necessary analysis, when both 
kinds of inertia are present, have been given in Walker’s 


830 Dr. J. W. Nicholson on the 


second paper, although the special case is not examined. 
Let e be the charge on the sphere of radius a, « its dielectric 
constant, and c’, ¢ the velocities of radiation inside and out, 
so that 


OO a oo eel or 


£ is the displacement of the sphere at time ¢, and F the 
force, of a mechanical origin. As for the conductor, the 
field outside can be expressed in terms of a function y (¢t—7), 
small like F’, in the forin, valid for a certain region, 


C 


r r é eé Ey e 
(X, Y, Z)= 3 (a, Ys 2) + 3(—1, 0, O)( 22x" + ry! +x */ 


+S (2, Ys 2)( 02x" + 30x + 3x3). sabia 


Inside the sphere, since there is no initial field, we may 
write, in terms of functions yy, (c't—r) and Wy, (c't+17), both 
of which are required, 


(X,Y, Z)= (1, 0, 0) {2° hi! + apo") + i! We!) + ity} 


+ 2, ys °C" +s") + 3rGh! —h') +3G ty) (8) 


the axes moving with the sphere. In order that the internal 
field may be finite at the centre, 


ri(e't) Habo(e't)=0. >.) es 


The tangential electric and electromagnetic forces are 
identical to the order contemplated, and thus by the con- 
tinuity of either at r=a, 


o( a?" = re, CMa de 2) =O! $a? (ry! + apa!) tal’ —Wo') +¥it wet. (5) 


The difference of normal flux being 47 or e/a’, it follows 
that 


e(ax’ Bi), fo is =Ke' (ary! —arre’+Wit+ye). . (6) 


C 


A determination of the mechanical force gives * for its 


* Walker, /. c. p. 174. 


Accelerated Motion of a Dielectric Sphere. 831 
resultant along the direction of motion the value 


2 ec , 
—3 ax (et—4) : 
so that 
ee CC rs 
mE + Bary 4 a a emt YD 
The initial conditions are as before, & —£=0 at i=0, 


x“(—a)=xy(—a)=0,. . . . - (8) 


and the equations for determination of (yw) subject to 
these conditions become 


ofatx' toy’ + (14 FF )— 3 

=o! fa? (ry! +o!) ta(hr'— ye) + hit yy} ; 
fay’ +(1+ = )y— 5a} 

=e {ay —We')+hit ype. - 2 6 © OY) 


and Walker’s particular solution for y involving only non- 
eae terms is 


(4 i 
mea) = = eK | ea 2maet oa (; mm 


ae m+m! m+m')? 


+ Gaga) fr OD 


where m'= a = and is the usual electrical inertia for slow 
speeds. 
The vibratory terms will be of the form 
y(ct—7) — Ap t—-7+a)/4 
Wi(ct—r)=Be~ —+aya +, |. (11) 


pa(clt-+r) = —Be itr +a)/a 


or more strictly, summations of this form for the various 
values of X 5 Ma the period equation 


emer pas(14 a) {e-n(t4 2-2) Bras} 


- (12) 
the real part of the summations being taken, A being obiiiies | 


RP > 
a 


832 Dr. J. W. Nicholson on the 


- The equation may be shown to have a root zero, but no 

others except complex values whose real part is positive. 

Thus the vibratory terms ultimately decay. ! 
We proceed to the case of a sphere without Newtonian 

mass. Taking the mass at first as very small, the non- 

y) 
a 3 = x becomes, on reduction, 
mE 


2m! c? 


vibratory part of 


K 
p—1 3” 


{ c7t? + Qact + 2a? 


and by (7) this is the non-vibratory part of m&. The 
vibratory portion is of the form 


mEDe—¥*l" sin (A +e), meray 


where the root of the period equation is now written A+ tm, 
and D and e depend on 2 and up. 

In order that & may satisfy the appropriate conditions at 
t=0, it is necessary that 


' i’ AW! iit 
=D sine=— alter hal ear 
2 
&(uD cose—AD sin e) = — a . oa 


the summation being for all roots X+c of the period 
equation 


tanh ka _ {i Kae? 


a KK +) 

The acceleration is always finite, whatever the distribution 
of the vibrations among the possible periods. The deter- 
mination of this distribution is difficult, but is not necessary 
for the present purpose. In addition to the decrease in 
amplitude which may be expected as the vibrations recede 
from the fundamental, there will be increased damping in 
the higher modes. When « is not too great, the damping 
will not be slight even for the fundamental, which will then 
be the only vibration needing attention. If this is so, and - 
if the amplitude of this vibration is sufficiently preponderant, 
we may write 


Z , t 
E= — (ee + 2act + oc) + De *@ sin (e + c), 


Accelerated Motion of a Dielectric Sphere. 833 


a‘ me tos ae 
where Dsne=— 75 ree 
a?7F XK 
pI) cos e= — oa(l— ~~), 
and for moderately large values of k, es fi, and 
= Hill 242 9 2 Fa’ —het/a 3, # 
oy 2m'c? (c*t! + 2act + 2a") — mep° sin (ct+a). (16) 


The period equation is practically tanhk?«=«?x, whose 
fundamental solution is «?v= +4°4932, so that w=4°493/x2 
of the order assumed above. For period equations of the 
present type, the real part of the solution is much smaller. 
A similar case is given by Lamb *. 

We see, therefore, that for a dielectric sphere under a 
small mechanical force, the vanishing of the Newtonian mass 
causes no difficulty as regards the acceleration ; and in view 
of the fact that the presence of this mass is doubtful, and 
that its absence would tend towards simplicity in the con- 
struction of the ideal electron, it seems possible that the 
postulation of dielectric rather than conducting property in 
an electron will be of service. 

Such an electron, moreover, by virtue of the rigidity of 
its electrification, would fulfil one of the necessary condi- 
tions for the validity of the quasi-stationary principle for 
small accelerations. It is the possibility of redistribution of 
the charge which is the main difficulty of this principle, and 
the problems treated by Walker are sufficient to show con- 
clusively that redistributions will ordinarily take place for 
conductors in accelerated motion. Now a fairly large value . 
of « for the dielectric interior of a sphere secures that the 
internal vectors shall be nearly zero, and this, combined 
with the rigidity of the charge, should be sufficient. It 
has been tacitly supposed throughout that the Lorentz 
contraction does not take place, although it is the belief of 
the writer that the contracted electron gives the best repre- 
sentation of fact, and a recent investigation by Bucherer f 
tends to prove this. 

If the dielectric sphere with a surface charge thus fulfils 
the conditions of that for which the quasi-stationary prin- 
ciple has been used, it may be expected to yield Abraham’s 
expression for the transverse inertia when the sphere has a 


* Camb. Phil. Trans., Stokes Commem. volume. 
1 Phys. Zeit. 1908, p. 775. 


834 Accelerated Motion of a Dielectric Sphere. 


uniform motion, and an accelerating force is applied per- 
pendicular to that motion. Now Walker has shown in the 
ease of the conductor, that when the surface condition is 
the evanescence of the tangential electromagnetic force, 
Abraham’s expression does not follow as the result of a 
direct calculation from the primary electromagnetic equa- 
tions. This disproves the quasi-stationary principle for the 
initial motions of a conductor at least, although the initial 
condition, involving the instantaneous creation of a uniform 
field, is somewhat artificial. 

The equations valid for a dielectric in variable motion are 
not yet free from doubt, and a direct calculation of the 
inertia in this case, as Walker points out, is not at present 
possible ; but he concludes that the inertia of the dielectric 
with a large value of « would be practically the same as for 
a conductor with equal charge, by the following argument *. 

“Since there is continuity of normal flux of disturbed 
electric force at the surface, the functions which determine 
the disturbance inside the sphere are of order «7! as com- 
pared with those which determine the outside field. Hence 
the tangential component of electric force inside, and there- 
fore also outside, is very nearly zero. Thus since the equa- 
tions for the ether are not modified by the motion of the 
sphere, the equation of motion and the surface forces outside 
differ by terms of order «7! from those for a perfect con- 
ductor. If this argument is valid, the assumption of perfect 
conduction, or of a high value of « for the charged particle, 
would equally well explain Kaufmann’s results, and give the 
same value for the electric inertia without limitation as to 
speed.” 

his argument appears to dispense with the necessity for 
complete analytical treatment. The inertia in question is that 
derived by Walker’s analysis of the conductor with the other, 
and in the opinion of the writer, less likely condition in that 
case, that the tangential electric force is zero. Quoting the 
results, the initial longitudinal inertia becomes 
e f4—50°+4 . ,, 4-134 6% ] 
[6ae hoses Tea 1G =e ov 
and the transverse inertia is 


e” AM? 24, hetifok tt ih Onley 
Se) FAS h+ pe Fee 


and these are the initial values to be regarded as true for a 
* Phil. Trans. 1910, A, p. 178. 


Electron Theory of the Optical Properties of Metals. 8395 


dielectric sphere whose coefficient is large. They do not 
agree with the results deduced from a consideration of steady 
motions, without redistribution, but must apparently be re- 
garded, with the corresponding values for a conductor, as the 
only values which have received a complete proof. 
Meanwhile, as stated in the preface, it may well be of service, 
in any attempt to treat the electron as not subject to defor- 
mation, to endow it with dielectric rather than conducting 
properties. The analysis of this hypothesis presents no difh- 
culty which does not appear to be shared by the other, and 
in a consideration of initial motions, it gives rise to great 
simplicity in the possible case of no Newtonian mass. 


XCI. The Electron Theory of the Optical Properties of 
Metals. By Prof. Harotp A. Witson, F.R.S., McGill 
University, Montreal*. of 


| ee electron theory of the optical properties of metals 
has been developed by Drude, J. J. Thomson, H. A. 
Lorentz, J. H. Jeans, and others. 

Let N denote the number of free electrons per c.c. in the 
metal, and let dN be the number in the group with velocities 
between V and V+dV. The number dN remains uearly 
constant, although particular electrons are continually enter- 
ing and leaving the group. Each such group may therefore 
be regarded as having a permanent existence. Since the 
mass of an atom is large compared with the mass of an 
electron, the velocity of an electron will not be much altered 
by collisions with atoms, and collisions with atoms must be 
much more numerous than collisions with electrons. Con- 
sequently the electrons in a particular group may be regarded 
as making many collisons, and still remaining in the same 
group or in a set of groups covering a small range of 
velocities. 

When an electrie force acts in the metal the electrons in 
each group will acquire an average velocity which will not 
be the same for the different groups. The motion of a group 
will be determined by a differential equation which will be 
of the same form for all the groups, but with different values 
of the constants for the different groups. It will therefore 
not be possible to represent the average velocity of all the N 
electrons by a single differential equation, unless we make 
the assumption that all the electrons have the same velocity 
of agitation. 

* Communicated by the Author. 


836 Prof. H. A. Wilson on the Electron Theory 


Let u denote the velocity of a particular group due to an 
electric force X acting parallel to the w axis and suppose 


(vide infra) 
: (dNmu) =dNXe—dNBu, 


where 8 is a function of V, m the mass, and e the charge of 
an electron. If Nuy= ‘ udN, up will be the average velocity 
of all the N electrons in the «x direction and 


3 (Nm) = NXe — { BudN. 


Jeans*, in his very interesting and valuable investigation, 
obtained the equation 


- (Nmuy)=N Xe—Neyuo, 


so that j BudN 
We j udN 


Since u varies with the time, it follows that y is not in 
general a constant. If, however, we take all the electrons to 
have the same velocity of agitation, we get y=. In Jeans’ 
investigation he took y to be independent ot the time, which 
seems to be equivalent to ignoring the velocity distribution. 
If this is not done, then it is necessary to find 8 as a function 
of V, which requires special assumptions to be made. 

Jeans obtained his equation on the assumption that the 
time dt in it is large compared with the time of a collision. 
If it is taken so large that during it a particular electron 
will have successively many velocities, then the velocity of 
all the electrons will be the same on the average over dé, and 
so the equation will be true. I think this requires dt to be 
large compared with the time between two collisions between 
one electron and other electrons (not atoms, since collisions 
with atoms do not alter the velocity much). If this is so, it 
means that Jeans’ equation will only be strictly correct for 
vibrations of much smaller frequency than those in infra- 
red radiation. Atthe same time, of course, it will be approxi- 
mately correct even for rapid vibrations, because the assumption 
that all the electrons have the same velocity of agitation 
represents the facts fairly well in such problems. 

In view of these considerations, it seemed to be worth while 
to work out the theory on the lines followed by Jeans, but 


* Phil. Mag. June 1902, and July 1909. 


of the Optical Properties of Metals. 837 


assuming that collisions do not alter the velocities of the 
electrons in magnitude. 

In the following paper the atoms are regarded as hard 
smooth spheres which remain at rest, while the electrons 
are also regarded as spheres and their mutual actions are 
neglected. These are the assumptions adopted by H. A. 
Lorentz* in his theory of the electrical and thermal con- 
ductivities of metals. 


Let n denote the number of atoms per c.c. in a metal and 
N the number of negative electrons, each with a charge e. 
Also let R be the sum of the radii of an atom and an electron. 
Let &, 7, € denote the velocity components of an electron and 
V its resultant velocity. Following Jeans, I shall begin by 
calculating the rate of increase in the momentum of the 
electrons due to an electric force X parallel to the # axis. 
Let 6N be the number of electrons per c.c. having velocity 
components between & 7, § and €+6&, n+6n, and €+ 66. 
Consider a particular atom and a small cone of solid angle 
dw with its vertex at the centre of this atom. Let f/ be the 
angle between the axis of this cone and the axis z. The 
number of electrons in the group ON which collide with the 
atom in time dt, so that at the moment of the collision the 
line joining the centres of the atom and electron lies inside 
dw, is 


SN R? do cos OVdt, 


where @ is the angle between V and the axis of the cone. 

The loss of momentum in the 2 direction at each such 
collision is 2» V cos 6 cos f, where m is the mass of an electron. 
Consequently the total loss of momentum by the group 6N 
is 


2mV? §N R? n dt | cos? 0 cos fda 
=2mV? ON R?n dt 7E[2V T 
=65N mEV dé[ln where Im=1/7n R?. 


Now let dN denote the number of electrons per c.c. which 
have velocities between V and V+dV, and let udN = { ESN 
so that w is the mean value of & for the electrons in the 
group dN. Then the loss of momentum by the electrons in 
this group is 

ve | EN= 


a 


mVudtdN 
bn ’ 


* “Theory of Electrons,’ pp. 266-273. 
t+ See H. A. Lorentz, ‘Theory of Electrons,’ p. 272. 


838 Prof. H. A. Wilson on the Electron Theory 


The gain of « momentum due to the electric force is 
Xe dN dt, so that we have 


NR ere 


™m 


a (mu dN)=XedN— 


Let now wp denote the mean value of & for all the N 
electrons so that Nuj= | udN ; then, if we find wu by means 
of equation (1), we can get uw) by putting the value of u in 


= | udN. 
‘We also have from (1) 


© (mNu)= XeN — =; 7 | Ved, 


s 

Comparing this with Jeans’ equation we get 
a \v dN 
a 


If we take all the electrons to have the same velocity V 
we get y=mV/Im. Jeans’ results consequently agree with 
mine only if all the electrons are taken to be moving with 
the same velocity. 

If X is constant then after a sufficient interval du/dé will 
be zero, so that w= Xelm/mV, and 


sesh ei es 


Up = NS ei dN. 


If we assume dN to be given by Maxwell’s law this gives 


j 3/2 > 1/2 
=o de (2) i Ve 'dV= 2( 4 ) pia ay 


Mm 


where g= 3/2V- 
Suppose now that X=a cos pt, then (1) gives 


ea cos ( pt—6) 
~ m(p + V7 {lm?) V2? 


where tan 6=pln [V. 


Hence again assuming Maxwell’s law 


g\2? ea (°* cos (pt —8) e—2V°V7dV . 
uy=4r(£) <{. (p PVA Re os) 


of the Optical Properties of Metals. 839 


The amount of heat produced per c.c. per sec. will be equal 
to the mean value of Neu,acos pt and to $a’a where o 
denotes the conductivity. Hence we get using (2) and 
taking the mean value 


“apy (Vn? Vera 
dae ic, ial fe J ee pe |? 


When p=0 this gives 


Hence 


J cen 
Py aeae ‘ L+p?m?o,.7/4qV?N7e rey 


If we take all the velocities to be equal we get instead 
o)=NelmimV, and 


bi G9 4 
o= 14 pmo? /N2e* athe wen 


which is the expression found by Jeans. Drude expressed 
o as a sum of terms like (4), one term for each class of 
electrons. If we regard each group dN as constituting a 
class, then Drude’s expression becomes identical with (3) 
allowing for the change of notation. Drude got it by 
assuming the motion of the electrons in each class to be 
opposed by a viscous resistance proportional to wu. 

Equation (4) has been used by Schuster and Jeans to get 
estimates of N from the values of o deduced from optical 
observations. The integral in (3) unfortunately cannot be 
expressed in finite terms, but it can be found of course by 
graphical methods. I find that (3) gives values of N about 
double those given by (4) in most cases. 


The conductivity for any frequency can also be obtained 
in another way by calculating the heat produced directly. 
Suppose as before that the metal contains N free electrons 
per c.c., and let / denote the length of a free path and V the 
velocity of an electron. The kinetic energy of an electron 
will be altered during a free path by the action of the electric 
force. It we calculate the total gain of kinetic energy for 
all the free paths traversed by all the N electrons in one 
second, the result will be the amount of heat energy produced. 


* H. A, Lorentz, in ‘Theory of Electrons, gets o, less by the 
factor 2/3. 


840 Prof. H. A. Wilson on the Electron Theory 


Let acospt denote the electric force so that mdvjdt 
=ae cos pt, where m is the mass, e the charge of an electron, 
and v its velocity component parallel to the electric force. 
Let ¢) denote the time at the beginning of a tree path and 
to +7 that at the end. Then 


M (Vp — Vy) = — {sin p(fo +7) —sin pt}, 


2 : 
i m(v,—%) = — cost; (to +7) sine 
Therefore 
2 
4m(v7?—v)") = = i: ) cos? s (2t) +7) sin 


le 3 
C (Qt) +7) sin 


ae 
+ 2v9— cos 
P 


If now S denotes a summation for all the free paths 
described by one electron in one second, we get 

1 fae . opr 

Sim (v7? — 097) =S— (“) sin? i 

because the mean value of cos? z is 4 and of cos#0, and we 


may make the addition in groups such that 7 is constant in 
each group. 

Suppose that the velocity V of a particular electron remains 
constant, then the number of free paths of length between / 
and 1+4di in one second is Ve~//""di/l2,, where Im denotes the 
mean free path. Alsol=Vzr. Hence 


 Lifae ye | La one ¥ 
Sim(v7? — v9") i) Si, sin av € dl 


a7 Lin 


~ ImV (1+ p22 | V2)’ 


The number of electrons for which V is between V and 
V + dV is, according to Maxwell’s law, 


3/2 f 
tnN(£) e 2’ V2dV, 


where g=3/2V’ and V’ is the mean value of V%. Hence 


of the Optical Properties of Metals. 841 
the heat H produced per c.c. per sec. is given by 
_ @rN@e*ln (q yi RUE Sache cihas ss 
Lae { SIE ane MN bik 


Hence 


2 3/2 
ees AnNe ao) ( 


co 
m \q e 0 


Ve-w'*dV 
as before. 
If 2 denotes the current carried by the dN electrons which 
have velocities between V and V+dV, then 1=eudN, which 

with equation (1) gives 

On Caner CN Re 
edaN dt Ls Lne7dN fo 
The theory of the propagation of light in the-metal follows 
from this equation in Drude’s manner*. If x denotes the 
refractive index and « the coefficient of absorption, we get | 


Ne?1,,” { My veer 


xX 


r2(L— 2) =1—167r/2g3? 


and n?x=2a/p. | 

If all the V’s are taken to be equal these equations reduce 
to the simpler form which applies when only one class of 
electrons is supposed present. 


The emission of light by a metal has been discussed by 
several writers. H. A. Lorentz has given a calculation of it 
on the assumptions adopted in this paper but applicable to 
very long waves only. J. J. Thomsonf worked out the 
emission for any wave-length on the assumptions that all the 
free paths occupy equal times and that the velocities of the 
electrons at the ends of each free path vary in a particular 
way. 

ions (loc. cit.) gives two calculations. In the first he 
assumes equal free path periods while in the second free 
paths are not assumed at all, but the calculation is based on 
the equation which seems to be exactly correct only when 
all the electrons have the same velocity. The following 
calculation is very similar to that given by H. A. Lorentz f, 
but it is not restricted to very long waves. It is therefore 


* ‘Theory of Optics,’ p. 397. 
+ ‘Corpuscular Theory of Matter,’ pp. 89-97. 
t ‘Theory of Electrons,’ pp. 81-90. 


Phil. Mag. 8.6 Vol. 20. No. 119. Nov. 1910. 3K 


. 842 Prof. H, A. Wilson on the Electron Theory 


only necessary to indicate where it differs from his and to 
give the final result. 

Lorentz shows that the amplitude in the radiation of a 
given frequency from a thin plate of the metal can be repre- 
sented in the form of a sum of terms; one term for each free 
path described by the electrons. Hach term contains the 


factor 
t+r 
{Ww cos —- . 7 (¢+2 )at, 


where 7 is the time of describing a free path. Lorentz 


assumes that the frequency is so small that cos <*t can be 


6 
regarded as constant during a free path, and so (page 87) 
puts this factor equal to 


TV~COS pe («+ *). 
6 c 


To make the calculation apply to greater frequencies it is 
only necessary to put the factor equal to 


ay. { sin = r(ttr+ 2 )—sin4 ee bee ~) 


and carry out the calculation in the same way as H. A. 
Lorentz gives it. 

In this way I get 
ee 3 A ub a7 


s 7 sin’ 


~ 264¢4r? 20° 
instead of Lorentz’s (page 88) 


se 
a= sa pigige © (): 
If the angle a is taken to be very small and 7?V.? put 


equal to 5 , the two expressions become identical as they 


should. 

The evaluation of S is similar to that given above in 
getting the heat produced in the metal. 

The final result for the perpendicular emissivity of the 
vlate of small thickness A is 


pieN Aln a V3e—2V"dV 
7 alee = pile e re 
which if p is taken to be very ein reduces to Lorentz’s 
expression. 


of the Optical Properties of Metals. 843 


If we take all the electrons to be moving with the same 
velocity V we get instead 


E Ne*ln p*VA 


~ 48er'e(1 + p2/V%) 


The coefficient of absorption A of a very thin plate is 
shown by Lorentz to be equal to ocA/c, where o denotes its 
conductivity. Hence using the value found for o when all 
the V’s are equal we get 
" Nel,A 


™m 


Hence we get for the energy density EH, per unit range of 
wave-length in full radiation, after putting mV’=2aT and 
p=2rer, 


Ea 87 E_ loral 
Shel aah A MERRO? 


which is the value found by H. A. Lorentz for very long 
wave-lengths, and also by Jeans. 

If the formule allowing for the distribution of V according 
to Maxwell’s law are used instead, we get 


” V3e—9V7*d V 
is pi 1+ pV 
3h OC” Veadv - 


( 


0 


It appears, therefore, that the extension of the calculation 
to shorter wave-lengths gives no indication of a diminution 
of E,. Both the emission and the absorption are diminished, 
but the diminution of the one compensates for that of the 
other. 

On the assumptions that the atoms are hard spheres which 
do not move and that the electrons do not collide with each 
other on which the above calculations are based, the electrons 
only gain or lose energy from the radiation. Consequently, 
those electrons which are moving with a greater velocity 
than the average will gradually lose energy, while those 
which are moving with a less velocity than the average will 
gradually acquire energy. After a time, therefore, all the 
electrons will acquire the same velocity, and the energy 
density will be given by 16zeT'/3\* exactly. The assump- 
tion of a velocity distribution given by Maxwell’s law is 


3K 2 


844 Mr. A. Stephenson on the 


therefore not really consistent with the other assumptions 
made. In reality of course the atoms do move to some 
extent and the electrons do influence each other, so that 
Maxwell’s law no doubt really holds good, but the absorption 
and emission are not exactly the same as those calculated. 
It seems likely that the absorption and emission calculated 
should be such as to make E,x=167reT/3\4 exactly. The varia- 
tion from this obtained above when Maxwell’s law is assumed 
is no doubt due to the neglect of the motion of the atoms and 
the mutual influence of the electrons. 

The value E,=167r2T/3A* is obtained exactly when the 
assumptions made and their consequences are strictly adhered 
to throughout, that is when all the velocities are taken to be 
equal. If Maxwell’s law is assumed, then the mutual en- 
counters between the electrons should also be allowed for to 
obtain an exact result. 

The equation H, = 167raT/3A4 has been very fully discussed 
by Jeans (loc. cit.), together with reasons why it fails in 
practice for short waves. It now seems very probable that 
the electron theory in its present form cannot account for 
the observed values of Hy. The observed values fall below 
those calculated for wave-lengths which are very large com- 
pared with those of Réntgen rays, which latter are still lon 
enough to be strongly absorbed by dense matter. Short 
ultra-violet light is even strongly absorbed by air. 


XCII. On the Intensity of Periodic Fields of Force. 
By ANDREW STEPHENSON*. 


13 hae nature of the motion of a system about a position 
of equilibrium in a periodic field of force depends 
upon the intensity of the field. In general, there is cumula- 
tive effect within each of a series of ranges of intensity, the 
period of the motion in the odd ranges being twice that of 
the field, and in the even ranges equal. 
If the field is an even function of the time, the equation of 
motion may be written. 


a 
£+2an? >, a, cosrnt.x2=0, 
0 
where a is regarded as a parameter measuring the intensity. 


* Communicated by the Author. 


Intensity of Periodic Fields of Force. 845 
The limits of the odd ranges are given by 


ui 
2a,+4,— 7 a, +a, a,+4, Ps Ah, (SR ey 8129 


9 


‘ . 
a,+a, 2a,+a,— 7 a,+a, 


| 
25 
a,+4a, a, +a, 2d, +o, — 7 


] 
2a,—a, — — a,.—a e—o braves er ue og Cee 
) 1 4a 1 2 2 3 ’ ( ) 
9 9 
a, — a, a,—a,— 7 4-4, 
9 25 
a, — a, Oi Oy ety Le 
and of the even ranges by 
9 3 0 eee 
2a,—a,— — =a, —4, a,— a, BANA == Wie TIAA nts CURES) 
Ce 2 ae 
a,—a, a,—a, 2a,—a,—— 
a, a, a, ° =0 e ° ° (iv.) 
I 
a, 2d, + a,— — a,+a, 
: 4 
/ a, a, +4, 24,+4,— a 


The motion within the ranges is of form 


AP OO) —WO) 5 + BePH P(t) +wt)}, « - C) 


where $(t) and W(t) are even and odd functions of t. 

2. For a simple periodic field, #+2an? cosnt.#=O0, the 
first three ranges of cumulative effect are 0°4542a}3°76; 
3°79 }2a>10°64 5 10°65$2a420°95. The relative smallness 
of the ranges of stability is noteworthy, 


846 Mr. A. Stephenson on the 


3. In the case of an alternating impulsive field 


%+2an? S cos (2r+ 1)nt.2=0. 
0 
2 


Ree: aN eg 4 é 
Hquation (i.) gives 2a2=1/ = On? =7 3, and (ii.) has 
no positive root. Hence there is instability if the impulsive 
spring is not less than four times the reciprocal of its period. 
This result may be obtained directly. If a point with one 
degree of freedom is subject to changes of velocity 8 x dis- 
placement and —@x displacement at intervals 7, then if the 
initial velocity is properly chosen relatively to the displace- 
ment, the displacements at intervals 27 increase or decrease 
in geometric progression provided 8t>2, the ratio being 


—2(A°r?—2+8r,/ p's —4). 
The displacements at the impulses are 
a, +kaa, —k'x, +h®ax, ba, ..., 
where 
k=}(Br+ V7 B'?—4) and a= V(6r—2)/(Br+ 2). 


The general solution is of form (1) where the period is 4r, 
and 


—$(27—t) =$(t) ={1—(1—afh,) t/t} eo" 
+4{1—(1+ak,)t/T}e-” between 0 and 7, 
p(27—t) =p (t) =41— (Lay) ef pe 
— {1—(1+ah)t/rbe-”* between 0 and 7, 


k, being the larger value of k, and p= loge k= : cosh7} Br/2. 


In the limiting case 8t=2, the particular solutions are 


Aigi(d) and By tégi(t)—yil)§, 
where ¢; and yy, are even and odd functions of ¢ defined by 
— $;(27—t)=¢,(¢) =1—1t/7 between 0 and 7, 
(27 — t) = Wy (t) =4¢(3—2t/7) between 0 and r. 


These give motions in which the displacements at the 
impulses are 6,0,—06, 0,5... and 0, —c,—4e, c, 8¢,... , results 
which are easily verified. | 

The case in which the impulses are of the same sign may 
be treated similarly, Since in the above the steady motion at | 
the limit of stability is unaffected by the omission of the 


Intensity of Periodic Fields of Force. 847 


negative portions of the spring, the condition for stability is, 
as before, that the impulsive spring be less than four times 
the reciprocal of its period. 

4, When the period of the field is made up of intervals of 
constant strengths, the periodic motions which distinguish 
the limits of the ranges may be found directly. I, for 
example, the spring is alternately equal to («n)? and —(an)? 
during intervals 7/n, the limits of the odd ranges are given 
by 

aah tan T (2a—1), eC — fan p2a—1), 


and of the even ranges by 
a 7 —a 7 
—e"™ = tan £ (2a—1), —e~?"= tan i (2a—1). 


The ranges of cumulative effect lie about the integral 
values of «, and those of stability about the values differing 
from an integer by a half; the latter become small without 
limit as « is taken large. In the case of instability the value 
of p, equation (1), is 

n cos (2e+ a7) 
dr 92° Gos (2e—az)’ 
where 
etar +] 
cos 2e=sin at Sah 

When the spring is alternately (an)? and zero for periods 
an, the ranges are given by cot a7 /2=a7/2 and cos a7/2=0, 
the even by tanaz/2=—am/2 and sin a7/2=0. Here the 
integral values of « give the upper limits of the ranges. 

5. The problem under consideration is of importance in 
connexion with those motions which are maintained by a 
periodic field of force. Take, for example, the ‘phonic 
wheel,’ in which “the essential feature is the approximate 
closing of the magnetic circuit of an electromagnet, fed with 
an intermittent current, by one or more soft iron armatures 
carried by the wheel and disposed symmetrically round the 
circumference ”*. ‘In some cases the oscillations of the 
motion about the phase into which it should settle down are 
very persistent and interfere with the application of the 
instrument. A remedy may be found in a ring containing 
water, or mercury, revolving concentrically.” 


* Rayleigh, ‘Sound,’ i. § 63. 


848 The Intensity of Periodic Fields of Force. 


The methods of ensuring steadiness and their significance 
may be arrived at theoretically. We assume that there is no 
independent driving, and that the field is of impulsive type, 
being constant through an interval which is small compared 
with the period. The equation of motion is 


6+ 20+ (nt) S 2,41 Sin (27+1) = G==0g 
0 


where ¢ is the angle between consecutive positions of stable 
equilibrium, and f(né) is an even periodic function of period 
27/n, equal to c? between 0 and 7, and zero between 7 
and an. 


2 Py 
Putting 7 @=nt—at aay we have 
€ 
b+ 2nd + 5: {(2r + 1)a2,41 cos (2r+1)a} f(nt) .d 


= & faz,41s8in (27 +1) a} f (nt) —Kn oe 
where ¢ is small. The mean value of the right-hand side 
is zero if 2 da,41 sin(2r+1)a=xel2c*r, which therefore 
gives the phase of the steady motion. The right now re- 
duces to the even function equal to xce/2r between 0 and r 
and —nkxe/27 between 7 and awj/n; this gives the small 
periodic variation in the angular velocity due to the inter- 
mittency of the force. The steadiness of the motion depends 
upon the free oscillation of ¢. From the concluding para- 
graph of §3 it is evident that there is cumulative effect if 
the impulsive spring is of sufficient magnitude. For a given 
phase, «, stability is assured either by the introduction of 


frictional resistance, proportional to 0, to absorb the energy 
communicated, as in the device cited above, or by a reduction 
in the spring to destroy the isochronism. The latter result 
may be attained by an increase in the radius of gyration. 

For a specified motional resistance and strength of current 
there is a limit of frequency below which there is instability, 
and the greater the radius of gyration the slower the speed 
at which the instrument can be used. 

If the field is not impulsive, but acts during an appreciable 
interval, ¢. g. half the period, steadiness might be reached 
in any case of instability, simply by a suitable increase of 
current. 


July 1910. 


[ 849 ] 


XCIII. Absorption and izeefesion of the B-Particles by Matter. 
By Atots F. Kovarixz, Ph.D.* 


ECENTLY considerable discussion has taken place on 
the law of absorption of the @-particles. Hahn and 
Meitnert have shown that the 8 radiations from radioactive 
elements are absorbed according to an exponential law, and 
according to their theory, namely, that radioactive elements 
emit rays of only one velocity, have concluded that such 
radiations are homogeneous. W. Wilsont has shown that 
the 8 rays separated out by a magnetic field, and therefore 
consisting of practically one velocity, are not absorbed ac- 
cording to an exponential law. The results of Schmidt §, 
~ Crowther ||, and others have confirmed in some respects the 
experiments of one or the other. The question whether the 
A-particles from radioactive elements are homogeneous or 
complex is still an open one, and obviously quite complicated, 
involving in its solution several other questions, such as the 
effect of the reflected radiations and the variation of the 
ionizing power of the 8-particle with its velocity, and possibly 
also with the path traversed. 

Since the 8 radiations from most of the radioactive materials 
are absorbed very nearly according to an exponential law, 
this law becomes very convenient in determining the co- 
efficients of absorption. The absorption curves are, however, 
influenced by different experimental arrangements, chietly 
because under some conditions the scattered radiation is 
not properly taken into account in the ordinary methods of 
measurement. 

The purpose of this investigation was (1) to determine the 
coefficients of absorption of the #-particles of different 
velocities under the most normal conditions; (2) to determine 
the effect of reflected rays on the coefficients of absorption ; 
(3) to determine the variation of the amount of reflexion 
with the velocity as well as with the reflecting substance. 
In order to do this, it is necessary to have sources of §8-par- 
ticles of different velocities, and for this purpose the following 
were employed :— 

Radium D+ which emits some weak rays probably from 
RaD but mainly the rays from Rak, whose coefficient of 
absorption by aluminium is 43°3 (cms.)~1, corresponding to a 


* Communicated by Prof. E. Rutherford, F.R.S. 

+ Hahn and Meitner, Phys. Zeit. ix. x. p. 321 (1908). 

t W. Wilson, Proc. Roy. Soc., A. lxxxii. p. 612 (1909). 

5) ..N¥. Schmidt, Jahrbuch d. Rad, u. Elek, iv. 4 (20), p. 451 eee 
|| J. A. Crowther, Proc. Camb. Phil. Soc. vol. xv. p. 442 (1910). 


ee eT ee — 


850 Dr, A. F. Kovarik on the Absorption and 


velocity represented by Hp of about 1720, where H is strength 
of field and p radius of curvature of rays. On account of 
the long period of RaD this source is constant, and hence 
very convenient. 

Actinium ©, which emits rays whose coefticient of absorp- 
tion by aluminium is 28°5, corresponding to a velocity repre- 
sented by Hp of about 2150. This was obtained by the recoil 
method for some experiments and by the use of actinium 
active deposit for others. 

Radium B, which emits rays whose coefficient of absorption 
by aluminium is 75°0, corresponding to a velocity represented 
by Hp of about 1200. This was obtained by the recoil method 
from radium A. 

Thorium A+B+C+D, the thorium active deposit, which 
emits rays of at least two velocities whose coefficients of 
absorption by aluminium are 110 and 16:3, which would be 
represented by Hp of about 900 and 2650, respectively. 

Radium active deposit whose rays bave a very wide range 
of velocities. 

With all the sources used except Ra D+H, the work is 
quite laborious on account of the corrections for the decay 
since the periods of all of the others are quite short. 

Generally, when the coefficients of absorption are obtained, 
the material is deposited on metals of considerable thickness, 
It will be shown in this paper that under such conditions 
the reflected rays play an important part in the coefficients 
of absorption. When thick layers of radioactive material 
are used, the B-particles from the various depths emerge with 
various velocities. In order to avoid these complications, the 
active material used was always in the form of a very thin 
layer deposited on a very thin aluminium leaf, whose absorp- 
tion or scattering effects were negligible. 


Scattering. 

The 8-particles from a uniform thin layer of radioactive 
material radiate equally in all directions. The measurement 
of the absorption by thin sheets is generally carried out by 
placing the radioactive matter some distance below an 
ordinary §-ray electroscope, and by placing the absorbing 
sheets at some distance above the active matter, so that only 
the more or less parallel rays normal to the sheets are con- 
sidered. If the absorbing matter is some distance below the 
opening in the electroscope, the scattering observed by 
Crowther*™ produces a steeper incline in the initial portion 
of the absorption curve than would be expected from the 
latter portion of the curve. If, however, the thin absorbing 

* J. A. Crowther, Proc. Roy. Soc., A. vel. lxxx. p. 186 (1908). 


Reflexion of the B-Particles by Matter. 851 


foils are placed directly on top of the thin radioactive material, 
and this is placed at some distance below the electroscope, 
an entirely different initial portion of the curve is obtained. 
This is clearly shown by the following experiments. Table I. 
and the corresponding curves in fig. 1 show the results. 


TABLE I, 
Ionization. Ionization. 
emery | | foul 
in mms. equivalent in 
Ra E. | Act O. Th A+D. mms. of Al. | Ra E.| Act C. Th A+D. 

0:0000 100°0| 100°0 100:0 0:00 100:0| 100°0 100-0 
0025 1000} 101°3 100°1 04 116°3| 121°8 120°3 
0050 100°3 | ‘i. ie ‘08 1060} 120°5 119°8 
010 100°8 | 103°0 105°8 a 92:0} 114:0 114:9 
020 103°3 | 105°0 105°9 16 79:9} 105-4 109°1 
030 103°7 | 105°5 106°0 ‘20 ee 95°7 1069 
035 BOBS lv dos me 24 i. 91°8 100:0 
040 uss 1069 107°0 28 ae 5) 94:4 
045 soo sae 32 A (a2 ae 
050 Be 107°4 109°8 ‘40 2h 58°8 82:0 
059 TOUS Nt ses =e ye se acs 67'4 
090 97°1| 105°5 101°6 64 oe ahs 78 
"120 87°5| 99°0 99°4 
150 SoA ei ee 
Lit LEE ay ee ek 
236 ins 75:0 90°6 
295 eet Pies i 
354 ois 56'1 79°4 
413 30°8 


fe 
BCCCERCERSE 
Bes alin! i=l eo NL Pips | SL 


Pit Ae Gs 
PEER EE Dis “ 


/ONISATION. 


‘010 §=020 §=-030 ‘010 §6-02c0 «6030060 010 020 )3=— 050 = -040 
iterate OF ALUMINIUM IN CMS. _ 


852 Dr. A. F. Kovarik on the Absorption and 


The first column gives the thickness of the absorbing 
layers of aluminium placed directly on top of the thin active 
layer. The second, third, and fourth columns give the ioni- 
zation observed in the electroscope. The ionization, when 
no absorbing matter was on top of the active material, is 
taken as 100. The remaining columns give similar results 
when tinfoil was used for which the equivalent absorbing 
thickness of aluminium is given. This equivalent value of 
tinfoil in terms of aluminium was found from the latter 
portions of the absorption curves, where, as will be seen 
from later work, the absorption curves are similar in form. 
Several radioactive materials were used in this determination 
with very similar results. 

It should be stated here that the opening of the electro- 
scope was covered with a considerable thickness of mica and 
tinfoil, so that the very easily absorbed radiations are not 
effective. It will be noticed that the initial portions of the 
curves rise to a maximum before absorption becomes at all 
obvious. Furthermore, the percentage increase is greater 
when tinfoil is used instead of aluminium. For a given 
source of 8 rays, the maximum is reached for equivalent 
thicknesses of tinfoil and aluminium, but this maximum 
shifts to the right on using more penetrating Brays. The 
observed effect is therefore a function of both the absorbing 
material and the velocity of the 8-particles. When the 
absorbing foils were placed at some distance above the active 
material, the maximum decreased in magnitude with the 
increase of the distance until finally the reverse effect ob- 
served by Crowther showed itself. By placing a perforated 
diaphragm above the active matter the maximum decreased 
in magnitude with the decrease of the size of the opening. 

These experiments show that the initial rise to a maximum 
is undoubtedly due to the scattering of the §-particles by 
the thin absorbing foils. In order that scattering may be 
complete, the @-particles must pass through a definite thick- 
ness of the absorbent. In the experiments described the 
radiation is equal in all directions. Those particles which 
strike the absorbent normally are scattered less than the 
oblique ones, provided the absorbing layer has a thickness 
smaller than that required for complete scattering. Conse- 
quently, the oblique rays, which did not reach the electro- 
scope when no absorbing matter was placed over the active 
material, become scattered when a thin foil is so placed, the 
degree of scattering depending on the thickness traversed, 
and therefore on the obliquity of the rays. These scattered 
8-particles reach the electroscope in numbers sufficiently 


Oe 


Reflexion of the B-Particles by Matter. 853 


large to more than compensate for the loss suffered by the 
normal rays due to absorption, reflexion, and slight side 
scattering. 

When a thick layer of radioactive material was used, e. g. 
uranium oxide, the effect described was not observed owing 
to the fact that complete scattering took place within the 
material itself except for the uppermost layer. The radiation 
from the latter was relatively too weak to produce an 
observable effect. 

It is clear, therefore, that scattering may produce a rise 
in the initial portion of absorption curves as well as a steep 
incline, or no observable effect, depending entirely on the 
experimental arrangements. In order to obviate the effect 
of the scattering of the @-particles on the initial portion of 
the absorption curve, the absorbing screens should be placed 
directly against the opening of the electroscope, in which 
case none of the scattered @-particles will be lost. 


Apparatus. 


In the ordinary cylindrical or rectangular shaped ionization 
vessels, the @-particles do not have equal paths, and conse- 
quently do not produce equal numbers of ions within the 
vessel. or this reason it was decided to use a hemispherical 
ionization vessel in the following investigation. A copper 
hemisphere, 30 cms. in diameter, was used for this purpose, 
and an appropriate electrode and electroscope were constructed 
as shown in fig. 2. The bottom of the ionization vessel was 


Fig. 2. 


S S 


removable. The active material was in all cases deposited as 
a thin layer on a thin aluminium foil (0:00025 cm.), and 


$54 Dr. A. F. Kovarik on the Absorption and 


this was in turn attached to a sheet of mica or aluminium of 
a thickness slightly greater than would be necessary to stop 
all the «-rays which ‘might be emitted by the active material. 
This sheet was then placed inside the ionization vessel, active 
side downward and over the central opening of the cover, to 
which it was held by means of springs. The absorbing sheets 
were also placed on the inside and were also held in position 
by springs. The change of capacity produced by the slight 
elevation of a large number of absorbing sheets was found to 
be negligible in the large vessel. The capacity of the instru- 
ment was about 10 E.S. units. Care was taken to obtain 
saturation in all the experiments. 

With this arrangement the -particles have the radine of 
the vessel for their path, and all the scattered rays become 
effective. Since, however, the @-particles after passing through 
matter have their velocity shghtly decreased *, then, if the 
ionization changes considerably with the velocity, the oblique 
rays will be affected more than the normal rays. In addition, 
with plane absorbing sheets, the oblique rays are more 
absorbed than the normal rays, and this should result in a 
slight drop in the initial portion of the absorption curve. 
Experiments were tried with more or less normal rays, but 
the absorption curves were nearly identical with those obtained 
when radiations in all directions were used. 

In some of the experiments, the interior of the vessel was 
lined with a thick cardboard covered with a conducting 
paper. The result was a decrease of about 20 per cent. of 
the ionization due to the fact that multiple reflexion of the 
8-particles is less from cardboard than from copper. 


Lifect of reflected B-particles on the absorption coefficient. 


The present form of the apparatus was well adapted for 
the investigation of the amount of reflexion* of the 8-particles 
by any substance, and also for the investigation of the effect 
of the reflected §-particles t on the absorption coefficient. 
When air was underneath the thin aluminium foil on which 
the active matter was deposited, the coefficient of absorption 
was always found to be smaller than when a reflecting sub- 
stance was placed under, and it increased with the atomic 


* W. Wilson, loc. cit. 

+ The terms “reflexion” and “ reflected B-particles or rays” are used 
for convenience only. By “reflected 8-particles”’ is meant the 8-particles 
which entering the substance, conveniently called the “reflector,” are 
deflected in their course by collision with the atoms of the substance so 
that they emerge again as diffusely scattered 8-particles. 


Reflexion of the B-Particles by Matter. 855 


weight of the reflector. When a differential curve was 
plotted, which would show the absorption of the reflected 
rays alone, the coefficient of absorption for the reflected rays 
was considerably greater than for the incident rays. For 
example, in the case of the 8 rays from Ra H, when air was 
underneath, the coefficient of absorption w=42°4 (cms.)—1; 
when lead was underneath » = 44°8, while the differential curve 
gave for the reflected rays w=51'0 cms. for aluminium. 
Schmidt* had already drawn attention to this fact. 

Table II. gives the values of the coefficients of absorption 
in terms of ems. of Al for the @-particles from Ra EH, ActC, 
Th D, and Ra C, when different substances are placed under 
the active material. 


TABLE II. 
Coefficients of Absorption. 
Substance 
underneath. 
Ra E. Act C. Th D. Ra C. 
51s 2a 42°4 27°6 15°% 13:0 
SLSR Sees 43°0 28°3 161 13°2 
a) CE A See ae ae oe 43°3 28'5 16:3 13°4 
ene 43°8 28°9 16°4 13°6 
eM tote ci 44:3 29°2 16°4 136 
i ES OER Bees Of 44°3 29°3 13:9 
a as ee 44°4 29°3 16°4 13°8 
nS eae 44-4 29°5 14:0 
2 A na ran ge 446 30°1 14:2 
[oe aie | 44:7 30°2 14:3 
1 2 ee 44:8 30°4 14-7 
ee SE | 448 30°4 16°7 14:7 
et RY, 44'8 30°4 16°7 14:7 
lt Ds ae eee’ 44:8 30°4 14:7 


It will be observed that in all cases the absorption coefficients 
are greater when a substance of greater atomic weight is 
placed underneath. The value for air underneath would 
apparently be the value of w for the incident rays. It follows 
from this that the average velocity of the §-particles is 
decreased by reflexion. 

Since the §-particles are reflected by matter, it is obvious 
that the absorbing matter above the active material will 
reflect downwards a certain fraction of the incident radiation 
the amount of which will depend on the velocity of the rays 
themselves (as will be shown later) and on the thickness and 


* H. W. Schmidt, Joc. c7t. 


856 Dr. A. I°. Kovarik on the Absorption and 


the atomic weight of the absorbing element. If air is under- 
neath, the 8-particles reflected by the absorbing sheets do not 
re-enter the ionization vessel in appreciable amount. Asa 
result, the initial portion of the absorption curve under these 
conditions will suffer a drop not due to absorption alone but 
due to reflexion as well. This effect is quite noticeable, as 
may be seen from Table III. and the curves 6 and ¢ in fig. 3, 
where aluminium and tinfoil were used respectively as the 
absorbing matter, air being underneath the active material 
in both cases. Evidently, to obtain a more accurate absorp- 
tion curve, the same thickness of the material used for 
absorption should be placed under the thin radioactive layer 
as is placed above it, in which case the loss of the B-particles 
by reflexion by the absorbing layers is compensated by the 
reflexion into the ionization chamber by the layers underneath, 
Column a, Table III., and curve a, fig. 3 (p. 857), show the 
results when aluminium is so used for the absorption of the 


8-particles from Ra E. 


TasxeE III. 
Absorption of the 8-particles from Ra D +- E. 


a=Aluminium as absorbing substance ; same number of 
layers underneath as used for absorption. 

6= Aluminium as absorbing substance; air underneath. 

c=Tinfoil as absorbing substance; air underneath, 


Aluminium Tinfoil in 

: a. b. equivalent value _ C. 

oT aniay of mms. Al, 

0:059 1009 100-0 0:059 100-0 
118 65°9 60°7 ‘099 65°5 
qt 46'3 44°4 “189 49°4 
236 84°6 33'9 179 40:0 
205. 27, 25°4 ‘219 32°5 
854 19°4 19°3 "299 22°8 
‘472 116 11°55 *309 19°4 
‘590 6°70 6:97 "459 11°55 
‘708 4:10 4°51 599 7°24 
826 2°41 2°41 "744 oll 
944 1-52 1:60 °870 1:87 

1:062 0°80 0°84 ‘939 | 1:38. 


The coefficients of absorption of the @-particles from Ra D, 
Th A, RaB, Ra E, Act C, Th D, and RaC, obtained in this 


manner, are given in Table IV. 


Reflexion of the B-Particles by Matter. 857 
Fig. 3. 


Pee atta det ch li. 
iS SeEMnnmMEE 
ct ee 


/ON/ISATION 


> 
aS 


exact 
SSE 
SEE 


er “020 -050 -040 del --060 -070 080 100 10 20 
ALUMINIUM IN CMS, 
TABLE IV. 


Coefficients of Absorption in cms. of Al. 


Ra D (?). ThA. | RaB. | Rak. | ActO. | ThD. | Ra. 


SS ee ee 


130 1110 75°0 43°3 28°5 16°3 13°5 


The coefficient of absorption of the @-particles from RaB 
was determined by using RaB obtained initially pure by 
recoil from radium A, as well as by difference method, while 
in the case of ThA and Ra D (?) the difference method 
alone was used. Act C was also obtained by the recoil 
method, but in some determinations the actinium active 
deposit was used. 

When the logarithm of the ionization given in column a, 
Table III. is plotted against the thickness of aluminium we 
still observe an initial drop. This must be attributed to weak 
radiations possibly belonging to Ra D. 

In the case of Ra H, Act C, and Th D absorption curves, 
it is found that down to about 5 per cent. of the initial 


Pin. Mag. 8.6. Vol. 20,.No. 119. Now 1910, 3. 


858 Dr. A. F. Kovarik on the Absorption and 


activity, the logarithm of the ionization when plotted against 
the thickness of the absorbing material gives a straight line. 
The conclusion generally drawn from such a result is that 
the absorption follows an exponential law, and that the rays 
are homogeneous. ‘This, however, should be done with due 
caution, for the logarithm of the ionization changes slowly 
with the ionization, and a straight line would be obtained 
when the absorption follows the exponential law only ap- 
proximately. This is clearly illustrated when we consider 
the differential curve for the reflected rays, say from lead, in 
the case of any of the above mentioned radioactive materials. 
For RaE rays, for example, after the soft rays are absorbed 
we get a straight line with the value of #=42°4 when air is 
underneath, a straight line with a value of w=44°8 when 
lead is underneath, and a straight line for the differential 
curve with a value of #~=51:0 cms. Al. Now, the differ- 
ential curve is obtained by taking the differences of two 
exponentials, supposing they are such, and this difference 
cannot, therefore, be an exponential ; but on account of the 
comparatively small differences in the exponents the 
differential curve approximates to an exponential. 

It must further be remembered that the rays after passing 
through matter are scattered and their velocity is changed, 
and while the @-particles of one velocity predominate there 
are, however, 8-particles of smaller and greater velocities* 
present as well, and a distribution of velocities of this kind 
may be the one which is required for such an approximation 
to the exponential law as is generally obtained. - 


Retlexion of the B-particles of different velocities. 


It was shown above that the reflexion of the @-particles is 
important in the study of the absorption curves. McClellandt 
and Schmidt{t have shown that elements of higher atomic 
weight reflect a larger percentage of the incident @-particles 
than the elements of lower atomic weight. In the course of 
this investigation it was noticed that the @-particles from 
Ra HE and ActO were not reflected equally readily by the same 
substance. ‘This suggested that the velocity of the @-particles 
plays an important réle in the problem of reflected rays. 
Systematic experiments were therefore carried out to study 
the amount of reflexion of the A-particles from Ra E and the 

* W. Wilson, loc. cit. 

ane McClelland, Sci. Trans. Roy. Dublin Soc. ix. pts. 1 & 2, pp. 1 & 
: t H. W. Schmidt, Jahrbuch der Radioaktivitét und Elektronik, iv. 
p. 451 (1908). 


Reflexion of the B-Particles by Matter. 859 


active deposits from actinium, thorium, and radium emana- 
tions by various substances. The apparatus used was the 
same as the one used in the absorption experiments. The 
active material was deposited on a thin aluminium leaf 
(0:00025 cm.), and this was attached to a piece of mica or 
aluminium of sufficient thickness to stop all the a-particles 
and yet not diminish the @-ray activity to any great extent. 
The plate was then fastened on the inside of the cover of the 
ionization vessel while the reflectors were held by springs on 
the outside. Readings were taken by having air underneath 
the active material, and then by placing the reflecting sub- 
stance underneath. ‘The difference in the ionization in the 
two cases divided by the ionization when air was underneath 
gave the percentage of reflexion. The reflected rays being 
somewhat softer than the primary rays would, consequently, 
be absorbed more by the mica or the aluminium than the 
primary rays, and a knowledge of the values of uw (Table IT.) 
is necessary to correct for this difference. ‘The results of the 
investigation are given in Table V. (p. 860). 

The numbers in the first column for each kind of rays give 
the mean of a large number of observations under the actual 
conditions, the thickness of the mica or aluminium sheet 
being given, while the numbers in the second column are 
the corrected values. The reflector was of sufficient thick- 
ness to produce complete absorption. 

It will be noticed that the §-particles from the actinium 
active deposit are reflected in greater proportion than those 
from Ra H, and that in the case when Th D predominates 
over Th A the percentage of reflected @-particles is greater 
than in the case when ThA and ThD are of about equal 
importance, so far as ionization is concerned. The @-particles 
from Ra B show a still smaller percentage of reflexion than 
those trom RaE. In these experiments the radium B was 
obtained by recoil from radium A by exposure in a strong 
field for a few seconds to a large quantity of active deposit, 
and measurements were made as rapidly as possible. Since 
radium B changes quickly into radium C and the latter 
emits 8-particles of higher velocity, the percentage of re- 
flected rays was found to increase with time, 7. e. with the 
quantity of radium C present. 

These results show conclusively that for the S-particles 
whose coefficients of absorption lie between 75 and about 
20 ems.—' Al, the percentage of reflected rays increases with 
the decrease of the coefficient of absorption, 7.e. with increase 
of the velocity of the @-particles. When more absorbing 
aluminium was placed in the path of the rays from the active 


3.12 


860 


Dr. A. F. Kovarik on the Absorption and 


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Reflexion of the B-Particles by Matter. 861 


deposits of thorium or of radium emanation, the percentage 
of the reflected @-particles rose to a maximum but finally 
decreased. The results with the thorium active deposit are 
- given in Table VI. and shown graphically in fig. 4. 


TaBLE VI. 


Thorium active deposit. 


| Aluminium Percentage of 
in mus. reflected 8-particles. 

| 0-045 65°3 

"104 77:0 
163 796 

“222 80:0 

"340 | 760 

"399 | 74:0 

"458 | 730 

"635 | 71°0 


The first column gives the absorbing aluminium in mms., 
and the second column gives the percentage of reflected 
8-particles. 


© 
(=) 


GE REFLECTION 


FERCENTA 


‘020 ‘030 
THICKNESS OF ALUMINIUM /N CMS 


This decrease may be due to one of two possible factors, 
viz., either the softer reflected rays are absorbed more than 
is accounted for by the correction or the amount of reflexion, 
as measured by the ionization, increases with the velocity of 
the §-particles only up to a certain velocity, after which it 
decreases again. Since this investigation deals with the 
heterogeneous rays from radioactive bodies, and not with the 
pure rays of any one velocity, this point cannot be answered 
here definitely. Investigation of this special phase of the 
problem is, however, now in progress (see following paper. 


by Kovarik and W. Wilson). 


$62 Dr. A. F. Kovarik on the Absorption and 


Multiple Reflexion. 


Since the @-particles are reflected from a substance into 
which they penetrate, the reflected rays should in turn be 
reflected if a substance is placed in their path. Suppose we 
have two parallel plates of lead and §-particles of the type 
pair Ra EK, say, to be emitted from the lower plate. Let us 
also suppose the velocity after reflexion to remain constant, 
and consequently the percentage of reflexion and the ioniza- 
tion of the particles involved in multiple reflexion to be 
constant also. The following calculation then shows what 
the ionization due to multiple reflexion should be. 

Let I represent the ionization due to the incident rays and 
p represent the percentage of reflexion. Then for the rays 
passing upwards the ionization is equal to 


I+pl+p?1+p°I+ ... &e. 


and for the rays initially passing downwards, after reflexion 


into the ionization chamber 
pl+p*l+p*I+ ... Ke. 
giving a total ionization of 


I+2pl(1+p+p?+p'+ ... &e.) 


a 


=587 per cent. of the ionization due to the incident rays; 
or that due to multiple reflexion alone is 487 per cent. 
The numbers substituted are obtained from Table V. 

Since the reflected particles have a smaller velocity than 
the incident, and the amount of reflexion decreases with 
decreasing velocity, it follows that the above ratio must 
be too high, unless the variation of the ionization with the 
velocity is very marked. In order to test this point and 
to see how the multiple reflexion varies with different 
reflectors and with rays of different velocities, a special 
apparatus was constructed. It was then found experi- 
mentally for the RaE radiation that the multiple reflexion 
was 236 per cent. of the incident radiation, measured by 
the ionization. This seems to justify the conclusion that 
the rays after the first reflexion are not as readily reflected. 
To see what the average value of the reflexion percentage 


es a 


Reflexion of the B-Particles by Matter. 863 


is after the first reflexion, we have, calling « the percentage 
of reflected rays after the first, the series 


I+ 2pl+2pa1+ 2pe7I+ ... &e. 
=I[1+2p(.+a+a?+a°+ ... &e.)] 


=I{1+ 20 


l—«z 3 


or 


336=100{ 14 <0", 
1—« 
from which «=40 per cent. 

Hence the percentage of reflected rays drops from 70:9 for 
the first reflexion to an average value of 40 for the following 
reflexions. This indicates that the slower # rays are less 
easily reflected than the swifter, unless the variation of the 
ionization with the velocity is large. 

The apparatus used in the multiple reflexion experiments 
consisted virtually of two ionization chambers. The parallel 


plates A and B (fig. 5) were connected together and to a 


70 BATTERY EARTH 


source of potential and had equal size openings a and 6 of 
4 cms. diameter; C was another plate with a little larger 
opening c and was separated from A and B by means of 
sulphur. C acted as an electrode, and was connected to a 
quadrant electrometer. a and 6 were covered with very thin 
aluminium foils so that the @-particles could readily pass 
through them without suffering much in absorption or re- 
flexion. The active material attached to a piece of mica or 
aluminium to absorb the e-rays was placed atc. The distance 
between A and B was varied from 1 mm. to 1 em. in order 
to see if the change of the solid angle of the issuing rays 
within the present size vessel produced variations in the 
multiple reflexion. Between 1 mm. and 3 mm. the readings 
were concordant, but with a greater distance the ratio for 
the multiple reflexion to the initial ionization decreased. 
Hence the smaller distances were used in these experiments. 


864 Dr. A. F. Kovarik on the Absorption and 


Readings were taken by measuring the ionization produced 
(1) when the rays from ¢ were allowed to pass through a 
and 6, (2) when the rays were reflected by the reflector 
placed over a, (3) when the rays below c were reflected 
upwards by placing the reflecting substance under c, (4) when 
the rays were multiply reflected by placing the reflectors 
over a and under ec. 

When the substance was over a the incident rays were 
reflected downward in a diffuse manner, and striking the 
aluminium (‘0059 cm.) over the active material become to 
some extent again reflected, &c. The readings in (2) were 
therefore always somewhat higher than in (3), which case 
(except for slight multiple reflexion from the interior of the 
chamber, amounting to about 3 per cent.) corresponds to the 
case when reflexion was studied with the hemispherical 
ionization vessel. Correcting (2) for reflexion and (3) for 
the absorption and slight multiple reflexion, the two readings 
always agreed. 

The results of these experiments for the rays from Ra H, 
using various substances as reflectors, are given in Table VII. 


TABLE VII. 


Reflectin Initial Ionization: Ionization :| Ionization: | Multiple 

rasta Po ioniza- reflector | reflector reflector on reflexion 
; tion. ontop. | under, | top and below.| alone. 
35 ES eee 100 182°0 172°6 336°0 236°0 
EM ie (ta heath f les | OO 337°5 237°5 
OE Ree ss 173°0 170°5 3130 213°0 
| | eee ernie : 173°4 156'8 79:0 179-0 
CHE sisdeesdueys. s 1575 | 1608 236°0 136 0 
Es 3650, 020 m 160°0 149°6 233°0 1330 
TE Sioa th idund . 154°5 141-4 224-2 124-2 
Tea three ig rs 150°8 1470 2180 118:0 
Ph tat gwecwncen a: 1432 135°7 189°0 89:0 
ANT ce ieuen “a 138°0 131°5 171-7 vb nt 
Ph hth hake - 121°5 121-0 146°0 46:0 
Cardboard... a 122°5 119°0 141°2 41:2 


The first column gives the substance used for reflecting 
the §-particles. In the following four columns are the 
values of the ionization obtained corresponding to the read- 
ings taken as given above, the ionization for the incident 
rays being taken as100. The last column gives the ionization 
due to multiple reflexion alone. 

It will be observed that the substances of higher atomic 
weight give larger values for the multiple reflexion (Pt for 


Reflexion of the B-Particles by Matter. 865 


some reason gives slightly higher value than Pb) than the 
substances of lower atomic weight. 

Experiments were next performed by using various radio- 
active substances as the sources of §-rays of different velo- 
cities but using lead in all cases as the reflector. The results 
are given in Table VIII. 


TaBLE VIII. 


Ionization due to multiple reflexion of @-particles of 
various penetrating powers. 


Th A+D. 


Ra E. ActC.| ThA=ThD. |ThD=90per ct..RaB+C.| Ra O. 


—— — 


236°0 250°0 | 74:0 225°0 2430 | 202:0 


The numbers in the various columns give the percentages 
of ionization due to multiple reflexion alone. The multiple 
reflexion increased in value for the more penetrating rays, 
but apparently reaches a maximum and then begins to 
decrease again. This agrees with the previous observations. 


Conclusions. 


(1) Scattering of the @-particles may produce a rise in the 
initial portion of the absorption curve under certain con- 
ditions, and a steep incline under other conditions. The 
effect of this scattering appears to be a function of the 
velocity. 

(2) In measurements of absorption the absorbing material 
should be placed directly against the opening of the electro- 
scope in order to avoid errors due to scattering. 

(3) The coefficient of absorption of the @-particles from 
thin layers varies with the substance underneath the active 
material, being greater for elements of greater atomic 
weight. 

(4) To avoid the steep incline in the initial portion of 
the absorption curve caused by reflexion of the -particles 
by the absorbing layers, it is necessary to place underneath 
the thin active layer as many absorbing sheets as above. 

(5) The percentage of reflexion, measured by the ionization 
method, of the §-particles from thin active layers by any 
one substance is a function of the velocity, and for rays whose 


$66 = Dr. Kovarik and Mr. Wilson on the Reflexion of 


coefficients of absorption by aluminium lie between 75 and 
about 20 (ems.)—! it is greater for the rays of greater 
velocity ; for the very penetrating rays the percentage 
decreases again. 

(6) The percentage of reflected 8-particles is greater for 
the reflectors of greater atomic weight. 

(7) By multiple reflexion the ionization may be increased 
from 100 to 350. The multiple reflexion changes in value 
with the atomic weight of the reflector, and the velocity of 
the @-particles in the same manner as the single reflexion. 

These experiments were suggested by Professor Rutherford, 
to whom I wish to express my deep gratitude for his valuable 
suggestions, attention, and the permission to work in his 
laboratory. 

Physical Laboratory, 


Victoria University of Manchester. 
July 4, 1910. } 


XCIV. On the Reflexion of Homogeneous B-Particles of 
Different Velocities. By Anois F. Kovariz, PhD., 
and W. Wuitson, J/.Sc.* 


N the preceding paper by one of us it was shown that the 
ratio of the number of 8-particles reflected from a sheet 
of matter to the number impinging upon it increased, within 
certain limits, with the penetrating power of the rays. This 
result was found for the heterogeneous rays emitted by 
radioactive bodies. The following experiments were made 
to determine the variation of this ratio for approximately 
homogeneous rays of different velocities. Such rays were 
sorted out by means of a magnetic field. 

The apparatus used is shown in fig. 1. The rays were 
emitted by the active deposit in equilibrium with radium 
emanation, corresponding to about 30 mmg. of radium, con-- 
tained in the bulb A. They entered the magnetic field B 
and described circular paths passing through the hole R 
and the tube § into an ionization vessel. The ionization 
vessel consisted of three parallel leaves of thin aluminium 
foil on rigid frames, 10 cms. x 10 cms.; the outer two metal- 
lically connected together were about one centimetre apart ; 
the inner one was separated from the outer two by means of 
sulphur, and it was connected by a protected wire to the 
gold-leaf system of a small electroscope E by means of which 
the ionization in the vessel was measured. The tube 8 was 


* Communicated by Prof, E. Rutherford, F.R.S. 


Homogeneous 8-Particles of Different Velocities. 867 


made of soft iron in order to protect the rays passing through 
it from the magnetic field, and thus prevent their deflexion 
to the sides. The ionization vessel rested on a plate P of 


Fig. 1 


the substance used for reflexion, which had a hole cut in it 
corresponding to the hole in 8. The plate P could be turned 
about the end D by means of an arrangement not shown in 
the figure, so that any desired angle with the horizontal could 
be obtained. A change in the inclination necessarily in- 
volved also a change of the tube 8. A plate of the substance 
used as reflector could be placed so as to cover the upper 
side of the ionization vessel as shown by Q. To determine 
the ionization by the y rays alone, a sheet of lead of sufficient 
thickness to absorb all the @-particles was placed between 
the pole-pieces of the magnet so as to cover the hole R. In 
the experiments with different reflecting substances the plates 
Pand Q were of the same substance. It was found necessary 
to keep the plate P always in position in order to reduce the 
effect of scattered radiation from various parts of the apparatus. 

During an experiment on the reflexion of the @-particles 
of any velocity the following readings were taken in the 
order given, and in the reverse order, repeating many times 
and using the mean values : 


(1) y-ray effect, Q in position ; 
(2) B+y-ray effect, Q in position ; 
(3) B+y-ray effect, () taken off ; ae 


(4) y-ray effect, Q taken off. 


868 Dr. Kovarik and Mr. Wilson on the Reflexion of 


Subtracting (1) from (2) we obtain the ionization due to 
multiple reflexion of the @-particles plus the initial ionization ; 
subtracting (4) from (3) we obtain the initial ionization of 
the incident 8 rays alone. The difference of these two values 
divided by the latter gives a ratio which is taken as a measure 
of the reflective power of the substance for the @-particles of 
the velocity under consideration. Owing to the fact that P 
had to be always in position and that some reflexion must 
occur in the ionization chamber even when Q is absent, this 
ratio is of necessity somewhat low, yet, the conditions being 
the same for rays of all velocities, the change of this ratio 
indicates the changes in the refiexion of the -particles of 
different velocities. 

Experiments were made with lead, copper, and carbon. 
The -particles considered were those between the values 
Hp=1164 and Hp=7660 gausscm. The results obtained 
when the angle of inclination was 45° are in Table I. and 
plotied in the curves of fig. 2. 


TABLE I.—lIonization vessel inclined 45° 1o the horizontal. 


| Ratio of the ionization due to 


multiple reflexion alone to the 
ionization of the incident rays. 
Hop in | rae "4 Pb 
gauss cm, | Cy (from curves), 
Pb Cu CO 

1164 | 0-56 0°25 0-02 2°56 
1372 0°94 abe diy 2°38 
1700 sad 0°53 0:10 2°25 
2072 1°36 seed 0:23 2:08 
2480 | 1°57 0°78 0:26 I‘91 
2920 | 1°60 0-77 0:27 2°04 
3400 1-64 bie ide 2:04 
3900 1°64 0-77 0°26 2°08 
4340 1°60 3 tie 2:05 
4720 1°58 0°77 0:25 2°10 
5000 1:56 des 0:22 2°12 
5380 1°53 0-71 nnd 2°12 
9720 a ae 0°18 ; 
6040 1°46 0°64 hy 2°12 
6320 1-42 ae ey 2°22 
6600 1°33 0:67 0:16 2°18 
6850 1°31 sins pad | 
7080 1°31 3 A 2°29 
7280 115 0°64 0-21 2°30 
7480 1:20 a ie 2°34 
7660 1:08 2'36 


2:18 mean 


Homogeneous B-Particles of Different Velocities. 869 


The first column gives the values of Hp. The following 
three columns give the ratio of the ionization produced by 
multiple reflexion to the ionization produced by the incident 
rays for lead, copper, and carbon respectively. The last 


FEFLECTION. 


0 900 ‘ 1800 2700 5600 4500 5400 6500 7200 8100: 


Hp in gauss cm. 


column gives the ratio of the values for lead and copper as 
obtained from the smooth curves. This ratio is roughly 
constant, indicating that the changes of reflexion with the 
velocity are proportional for the two substances. The ex- 
periments with carbon were difficult on account of its weak 
reflecting power, and the results are consequently subject to 
greater errors than in the case of the other two substances. 
The experiments with an angle of 30° for the inclination of 
the ionization vessel with the horizontal were tried with lead 
only, and the results were materially the same as for 45° 
inclination. For the very slowly and very rapidly moving 
8-particles the experimental errors are greater on account 
of the fact that the B-ray effect is small compared with 
the y-ray effect, since the distribution of the number of 
8-particles with the velocity gives a maximum and decreases 
quite rapidly on each side. This accounts for the irregu- 


_. larity of the observations at both extremities of the smooth 


curve. 

The results here obtained fully confirm the results of the 
investigation with the heterogeneous rays from radioactive 
bodies as given in the preceding paper. It will be observed 
that the diffusely reflected radiation as measured by the 
ionization rises rapidly to a maximum and then slowly begins. 
to fall. If the @-particles were reflected with the same 
velocity with which they impinge on the matter, or if the 


870 Messrs. Gray and Wilson on the Heterogeneity 0) 


ionization did not change materially with the velocity, then 
the rise and fall in the curve would have to be explained by 
supposing the number of -particles reflected to increase 
with the velocity up to a certain value and then decrease 
again. Since, however, the ionization produced by #-par- 
ticles of different velocities may play an important part in 
the results obtained, a satisfactory explanation of this pheno- 
menon cannot be arrived at until we know definitely the 
variation of the ionization due to §-particles moving with 
different velocities. 

In conclusion we wish to express our best thanks to 
Prof. Rutherford for suggesting this research. 


Physical Laboratory, 
Victoria University of Manchester. 
July 25, 1910. 
y ’ \ ; 


ah ——— — Se eee ee 


XOV. The [Heterogeneity of the B Rays from a Thick Layer 
of Radium FE. By J. A. Gray, B.Sc., 1851 Exhibition 
Scholar, Melbourne University, and W. Wiison, M.Sc., 


Hon. Research Fellow, Manchester University*. 


HE law of absorption of 8 rays by matter has lately 
been the cause of some discussion. Until the ex- 
periments of W. Wilson f, it had generally been assumed that 
8 rays absorbed exponentially by aluminium were homo- 
geneous. It was shown, however, that the coefficient of 
absorption of approximately homogeneous 8 rays rapidly 
increased with the thickness of matter traversed, which 
suggests that the rays experience a diminution in velocity 
as they pass through the aluminium. Crowther { obtained a 
similar result by the same method. 

The decrease in velocity indicated by these experiments 
has been determined directly §. It follows as a necessary 
consequence of these results that 8 rays which are absorbed 
exponentially by aluminium are not homogeneous. 


Recently, however, v. Baeyer and Hahn ||, using a photo- | 


graphic method. have shown that the 8 rays from several 
radioactive products initially possess a considerable degree of 
homogeneity. We have no definite evidence so far that the 


* Communicated by Prof. E. Rutherford, F.R.S. 

+ Wilson, Proc. Roy. Soc., A. lxxxii. 1909, p. 612. 
t Crowther, Proc. Camb. Phil. Soc. xv. pt. v. p. 442. 
§ Wilson, Proc. Rey. Soc., A. Ixxxiv. 1910, p. 141. 

|| v. Baeyer & Hahn, Phys. Zeit. xi. 1910, p. 488. 


the 8 Rays from a Thick Layer of Radium E. 871 


rays from such thin layers as they used are absorbed 
according to an exponential law. 

Gray * by the same method showed that the @ rays from 
a thick layer of radium E are distinctly heterogeneous, 
although they are absorbed according to an exponential law 
by aluminium. In view of the experiments of v. Baeyer and 
Hahn the following experiments were performed. 

The apparatus is shown in fig. 1 and is similar to that 
used by Wilson (loc. cit.). 


Fig. 1. 


The rays from the radium H which was placed at B could 
pass through a hole ina lead block A and into a magnetic 
field perpendicular to the plane of the diagram. They were 
bent round in circular paths and passed through holes O 
and P in lead screens into an electroscope EH. The mean 
radius of the path of the rays was 4 cm. and the diameters 
of the holes O and P*8 cm. The ionization in the electro- 
scope was determined for several strengths of the magnetic 
field, and the results obtained are shown in fig. 2a, where 
the ionization in the electroscope is plotted against the 
product of the field strength and the radius of curvature of 
the path of the rays. If the rays were homogeneous and 
the holes O and P and the source infinitely small, we should 
only get ionization in the electroscope for one definite 
strength of field. Experimentally however, even if the 


* Gray, Proc. Roy. Soc., A. lxxxiy. 1910, p. 136. 


872 Messrs. Gray and Wilson on the Heterogeneity of 


rays were homogeneous we should get a curve somewhat 
similar to that shown, on account of the necessarily finite 
size of the holes O and P. The following test of the 


Fig. 2. 


20 


JONISATION 1/N ARBITRARY UNITS. 


.000 6.000 
4p Nn GAUSS CM. 


homogeneity of the rays was therefore applied. A sheet of 
aluminium of known thickness was placed in the path of 
the rays just underneath the electroscope, which was about 
9 mm. above the hole P, and the experiment repeated. Now 
if the rays are heterogeneous, those which enter the electro- 
scope with the higher magnetic fields should be less easily 
absorbed than those which enter with the lower. The 
maximum point should therefore move to the higher fields 
when sheets of aluminium are placed under the electroscope. 
That this is the case is shown unmistakably by the curves 
of fig. 2, a, b, ¢, d, e, which are plotted from the results 
obtained with 0, ‘067, -245, 489, -731 mm. of aluminium 
respectively under the electroscope. If, on the other hand, 


the B Rays from a Thick Layer of Radium E. 873 


the rays were homogeneous there would be no such change, 
since for any strength of field the percentage change in 
ionization would be the same for the same absorbing screen. 

The change in the position of the maximum is quite 
marked, being at Hp = 2,400 in curve a, and at Hp = 3,650 
in curve e. 

It will be noticed that the rays which produced the 
maximum ionization when no aluminium was placed under 
the electroscope are practically all absorbed by a thickness 
of -73 mm. Al, while for rays corresponding to the higher 
fields appreciable quantities are still transmitted. 

In these experiments the bottom of the electroscope was 
closed by a sheet of tinfoil so that the maximum point even 
with no aluminium under the electroscope occurs at a rather 
higher field than it would if no matter whatever were placed 
in the path of the rays. An experiment was therefore made 
with a dutch-metal leaf closing the bottom of the electroscope, 
and the maximum point, which was carefully determined, was 
found to occur for a value of Hp 2,200 gauss cm. 

This value is in very good agreement with those obtained 
by Schmidt * 2,200 and Gray 2,300. Experiments made 
with tin as absorbing medium gave the same type of 
result. 

Absorption curves were taken for rays corresponding to 
various field strengths and are shown in fig. 3, curves a, } 
and d, in which the log of the ionization in arbitrary units is 
plotted against the thickness of absorbing material. The 
curves are for rays corresponding to Hp = 3,200, 2,400, and 
1620 gauss cm. respectively. It will be seen that the 
different pencils of rays have very different penetrating 
powers, the initial absorption coefficients being 13:0, 22:2, 
and 62°5 cm.~' for curves a, 6, and d respectively. 

It is especially significant that bundles of rays forming 
a large percentage of the whole can be separated out which 
have initial absorption coefficients much less than that of the 
whole beam. If homogeneous rays were absorbed according 
to an exponential law the absorption coefficient of the whole 
beam would continually decrease, which is not found to be 
the case. 

The source of radium E was a thick layer of radium D, 
so that any soft rays are practically all absorbed in the 
material. Asa further precaution the active material was 
covered with ‘1 mm. of aluminium. 


* Schmidt, Phys. Zeit. viii. 1907, p. 361. 
hil. Mag.8: 6. Vol. 20. Nor 419, Wove i910. 3 M 


874 Heterogeneity of 8 Rays from T hick Layer of Radium E. 


That the type of absorption curve obtained for homo- 
geneous rays was not due to the arrangement of apparatus 
was shown as follows. The radium E was placed 6 cms. 
below the hole P and an absorption curve of the whole 
bundle of rays taken. The result obtained is shown in 
fig. 3 curve c. The curve is exponential, showing that the 


JN ARBITRARY UNITS. 


Z0G OF /SONISATION 


“4. 
THICHNESS OF AL /N MMS. 


apparatus used has no effect on the absorption curves and 
the absorption coefficient found 42°8 cm.~' is in very good 
agreement with the results obtained by other observers. 
The initial drop is due to scattering of the rays and not to 
the presence of any soft radiation, since the same percentage 
drop is obtained if various thicknesses of aluminium are 
placed directly over the radium E and the absorption curve 
again taken. 


Deflexion by Electrostatic Field of Radium B. 875 


Conclusion. 


It has been shown above that from a pencil of @ rays 
which is absorbed by aluminium according to an exponential 
law, rays of widely different penetrating powers can be 
separated out. It follows, therefore, that absorption of 
8 rays according to an exponential law is no criterion of 
homogeneity. ‘The experiments confirm the results obtained 


by Gray by the photographic method. 


We wish to express our best thanks to Prof. Rutherford 
for the kind interest he has taken in this research. 
Physical Laboratories, 


The University, Manchester. 
Aug. 19th, 1910. 


XCOVI. The Deflexion by an Electrostatic Field of Radium B 
on Recoil from Radium A. By SIDNEY Bo DS: ang, 
Water Maxower, M.A., D.Sc.* | W 

Introduction. Cy 


N a previous paper an account was given of some 
attempts made to determine whether radium C is 
electrically charged when it recoils from radium B ; but the 
experiments made for this purpose failed to reveal any such 
charge. The formation of radium C from radium B is accom- 
panied only by the emission of B-particles, and it was thought 
that a transformation involving the expulsion of an «-particle 
might afford a more satisfactory case for investigation. 
Experiments were therefore undertaken on the recoil of 
radium B from radium A. In this transformation the atom 
of radium B should be negatively charged after recoil, if the 
process is accompanied simply by the expulsion of an 
a-particle, as is usually supposed. If, however, there is a 
simultaneous evolution of £-particles, the recoiling atom 
might be electrically neutral or even positively charged. 
It will be remembered that radium A on formation from the 
emanation in air at atmospheric pressure does in fact acquire 
a positive charge, so that it is concentrated on the negative 
electrode when the emanation is subjected to an electric 
field t. The following experiments were made with the 
object of determining whether the radium B is charged when 


* Communicated by the Authors, 
+ Makower and Russ, Phil, Mag. Jan. 1910. 
t Rutherford, Phil. Mae. Feb. 1900. 

3M 2 


876 Drs. Russ and Makower: Deflexion by Electrostatic 


i recoils and, if so, its sign; the magnitude of the deflexion 
of the recoiling atoms by an electric field has also been 
determined. A great number of experiments has been made 
by different methods, which show that the “ recoil-atoms ” * 
of radium Bare positively charged. The process of the form- 
ation of radium B from radium A would therefore seem to be 
accompanied by the expulsion of 8 rays as well as @ rays, 
though the speed of the former may be too small to allow of 
their detection by ordinary methods. 


Preliminary Experiments. 


In the earlier experiments, two insulated brass plates 
3°5 centimetres long and 1:7 centimetres wide were mounted 
at a distance of one millimetre apart inside a 
glass vessel which could be rapidly exhausted to ae 
a high vacuum. A platinum wire was exposed to C. 
radium emanation for about ten minutes to obtain 
as much radium A on it as possible and then trans- 
ferred to a tube which could be quickly exhausted, 
contained in a furnace at about 400° C. It was 
found that the emanation adhering to the wire could 
in this way be completely removed. The wire 
was then mounted at W, as shown in fig. 1,so AWB 
that the “recoil-stream” from it was projected 
symmetrically between the two plates A and B. 
An electric field could be applied by connecting 
the two plates respectively to the two terminals 
of a storage battery by leads sealed through the W 
glass vessel. 

After ten minutes’ exposure in vacuo the plates were 
removed from the vessel and the distribution of activity on 
each tested by mounting them in turn on a movable plat- 
form, and bringing successive strips of the plates under a 
rectangular window 3 centimetres long and 3 millimetres 
wide, cut in the base of an electroscope and closed by an 
aluminium leaf. The a and £8 radiation from the section 
of the plate just under the window could thus enter the 
electroscope, but the radiation from the rest of the plates 
was prevented by lead screens from contributing to the 
ionization inside the electroscope. From such measurements 
the number of “ recoil-atoms” projected from the active 
wire on to the different portions of the plates could be 
compared. A measure of those undeflected by the electric 


* We propose to use this term for brevity, to denote the matter which 
recoils as the result of a radioactive process. 


Field of Radium B on Recoil from Radium A. 877 


field was obtained by testing the cross-piece C in a similar 
manner. 

The results of a series of experiments made in this way 
with different voltages between the plates led us to believe 
that at least some of the “recoil-atoms” of radium B 
projected from the wire were positively charged. 

To test the correctness of this conclusion with certainty 
the following experiments were made. 


Direct Determination of the Quantity of Radium B deflected 
by an Klectric Field. 


The active wire W (fig. 2) was placed symmetrically between 
two pairs of plates 1:4 millimetres apart and 4 centimetres 
long adjusted to be in line with each other and contained in 
a glass vessel which could be evacuated as in the previous 


Fig. 2. 


A ee B 
Ww 

experiments. The “recoil-stream”’ from the wire W was thus 
projected through the gaps between the two pairs of plates, 
and fell upon the cross-pieces A and B. Between one pair of 
plates an electric field was maintained, while the other plates 
were kept at the same potential. If the “ recoil-stream” 
were charged on leaving the wire, it would be deflected by 
the electric field, and the plate B should therefore receive less 
activity than A. Since the radium A on the wire might not 
have been deposited uniformly by exposure to the emanation, 
the wire was kept constantly rotated by attaching it toa 
ground-glass stopper turned by hand and fitted into the 
containing vessel. After an exposure of ten minutes in 
vacuo to the radiation of the wire, the two cross-pieces A 
and B were removed and tested by an a-ray electroscope. 

The number of ' “recoil-atoms” reaching the plate A 


‘through the uncharged plates was always found to be greater 


than the number reaching B through the electric field. The 
reduction in the number reaching B depended on the field- 
strength, being greater the greater the field applied; but 
although the quantity of radium B reaching the cross-piece B 
was reduced in this way by the field between the plates, some 
activity was always found there even with the greatest field 
used, indicating that part of the “ recoil-stream” projected 
from the wire was undeflected. In view of some subsequent 
experiments on the magnetic deflexion of radium B it seems 


878 Drs. Russ and Makower: Dejlexion by Electrostatic 


probable that this undeflected portion was deposited on the 
strip during the exhaustion of the apparatus ; owing to the 
large electric fields used, it was not possible to apply the 
voltage until the evacuation was almost complete. The 
fraction of the ‘ recoil-atoms ” deflected by the electric field 
with different voltages between the plates can be seen from 


Table I. 


TABLE I. 
Voltages between | Activity on Activity on 
plates. | eross-piece A, cross-piece B. 

0 100 98 

340 | 100 69°5 

340 100 | 68°5 
1110 | 100 50 
1180 100 38 


Magnitude of the Electric Deflexton. 


Having established that the “recoil-stream” in a high 
vacuum is deviated by an electric field, it remained to 
determine the magnitude of the effect, to see whether the 
observed deflexion is in agreement with that calculated on 
the following simple assumptions. Since radium B is formed 
from radium after three successive stages, at each of which 
an @ particle is evolved, then since the atomic weight of 
radium is 226 and that of helium 3°96, the atomic weight of 
radium B should be 214 according to this view. Taking this 
value, then, if the velocity of the a particle from radium A is 
taken as 1:77 x 10° centimetres per second*, we obtain from 
the equation of momentum the value 3°27 x 10? centimetres 
per second for the velocity of the “ recoil-atoms ” of radium B. 
Assuming, further, that the atoms of radium B on recoil each 
carry with them 4°65x10~-? electrostatic unit (the charge 
carried by the hydrogen ion in electrolysis), it is easy to 
calculate the radius of curvature of the particles when passing 
through an electrostatic field applied at right angles to the 
direction in which they are travelling. 

A number of experiments made to measure the magnitude 
of the deflexion of the “recoil-atoms” of radium B in an 


* Rutherford, Phil. Mag. Oct. 1906. 


Field of Radium B on Recoil from Radium A. 879 


electric field gave inconclusive results ; for it was found that 
surfaces, even though situated so that they could receive no 
direct radiation from a source of radium A, became active. 
It is unnecessary here to enter into a discussion of the 
mechanism by which this occurs. 

It thus appeared that no reliable results could be obtained 
unless the possibility was excluded of much active matter 
reaching the receiver by methods other than direct radiation. 
The apparatus shown in fig. 3 was therefore designed to 
obviate this trouble. 


Fig. 3. 


<—_——— Sma $2 ens > 


The wire W was mounted between two parallel mica strips 
2 centimetres long, coated with copper plates kept at the same 
potential. The ‘ recoil-stream”’’ from the wire passed from 
the region between the two copper plates where no electric 
field existed into the space between the two parallel brass 
plates A and B, 9:4 millimetres long, between which a 
difference of potential was maintained. The plates A and B 
fitted into the ebonite plug E and were 1:17 millimetres 
apart. The copper cylinder K, which was 4°5 centimetres 
long and fitted over the ebonite plug EK, prevented the 
““yecoil-stream ” from being subjected to an electric field 
after leaving the space between the plates A and B. The 
“‘recoil-stream ” subsequently fell on a brass strip C, 2 centi- 
metres long, situated 5 millimetres from the end of the 
copper tube. 

To carry out an experiment the wire W was exposed to a 
large quantity of emanation for ten minutes, transferred to a 
vessel at 400° C. which could be quickly evacuated to remove 
adhering emanation, and then mounted as shown in fig. 3. 
The whole apparatus just described was contained in a glass 
tube which could be exhausted to a pressure of about 
1/300 millimetre of mercury within two minutes, after which 
the electric field could be applied between the plates A 
and B without fear of a discharge taking place through the 
residual gas inthetube. The difference of potential between 
A and B was determined by a direct-reading electrostatic 


880. Drs. Russ and Makower: Deflexion by Electrostatic 


voltmeter. After an exposure of ten minutes to the “‘ recoil- 
stream’ from the wire, the distribution of activity over the 
strip was tested in a manner similar to that described for 
the preliminary experiments, except that the width of the 
aluminium window in the base of the a-ray electroscope was 
reduced from 3 millimetres to 1 millimetre. 

The distribution of activity on the strip is shown in fig. 4 


Fig. 4. 


\qeld  \9s0Wolls = \ gro Vole 


5 2 ae ay oe ay a a oe aa a ae 


MtLhimehres 


for three typical cases when no field was applied between 
the plates and with 930 and 1920 volts respectively. To 
show that there was no lack of symmetry in the disposition 
of the apparatus, measurements were also made with the 
field reversed. The results so obtained were in substantial 
agreement with those shown in the diagram. 

It will be seen that with no field the activity of the strip 
exhibited a well-marked maximum at its middle, while the 
activity fell off rapidly and symmetrically on either side. 
With 930 volts the point of maximum activity was displaced 
2 millimetres, but the distribution was no longer symmetrical 
about the maximum. With 1920 volts the curve of dis- 
tribution showed two maxima, one near the centre of the 
strip and the other about 4 millimetres from the centre. 

It is not clear what is the cause of these two maxima, but » 
one maximum always occurred at the middle of the plate and 
was therefore due to “ recoil-atoms”’ which probably for the 
reason already suggested, reached the plate without being 


Field of Radium B on Recoil from Radium A. 881 


deflected. The distance between the two maxima was taken 
as a measure of the deflexion of the “‘recoil-stream” while 
passing through the electric field. Although this procedure 
does not give accurate values, it serves to show the order of 
magnitude of the deflexion. 


TABLE ITI. 
| 
Displacement of Displacement of 
eis a maximum observed, |maximum calculated, 
P ; in millimetres. in millimetres. 
930 2:0 1'62 
1680 39 2°94. 
1860 4-5 * 3°25 
1890 4:0 o31 
1920 4°] 3°36 


* In this case the readings were small and correspondingly more 
uncertain than in the other experiments. 


If the velocity of the “recoil-atoms” of radium B is 
assumed to be 3°27 x 10’ centimetres per second, as calculated 
on p. 878, it is possible to deduce the deflexion to be expected 
on certain simple assumptions. Tor the particles projected 
from the wire will have a parabolic path as they pass between 
the two charged plates, and will proceed along the tangent 
to the parabola at the point at which they emerge from the 
electric field until they ultimately strike the strip mounted 
to receive them. From the known dimensions of the 
apparatus it is a matter of no great difficulty to calculate 
the displacements to be expected on the strip for different 
voltages applied between the plates, assuming that the value 


of = for the hydrogen ion in electrolysis is 9°63 x 10° on 


the electromagnetic system, that the charge carried by the 
atom of radium B is the same as for the hydrogen ion in 
electrolysis, and that its mass is 214. 

There is, however, one source of uncertainty in making the 
calculation ; for since the length of the plates was only 
9-4 millimetres, which is not very great compared with their 
distance apart, which was 1:17 millimetres, the field must 
spread out appreciably at both ends. The corrections for the 
end effects in cases similar to that of our experiments have 


882 Dr. Makower and Mr. Evans: Deflexion by Magnetic 


been worked out by Coffin*, and it was estimated that on 
this account the effective length of the plates in our experi- 
ments was about four per cent greater than their actual 
length. In the calculation the value 9°8 millimetres has 
therefore been taken as the length of the plates instead of 
their real length, 9°4 millimetres, and the numbers given in | 
column 3 of Table II. were thus obtained. 

An inspection of the calculated and experimental deflexions 
shows them to be of the same order of magnitude, and we 
may therefore conclude that if radium B carries the unit 
charge of electricity, its atomic weight is of the order to be 
expected on the disintegration theory of radioactivity. — 

Our thanks are due to Professor Rutherford not only for 
supplying us with the radium emanation necessary for the 
experiments, but also for his interest in the work. 


XCVII. The Deflexion by a Magnetic Field of Radium B on 
Recoil from Radium A. By W. Maxower, M.A., D.Sce., 
and BH. J. Evans, B.Sc. 


[Plate XVIII. ] 


T has been shown by Russ and Makower { that radium B 
is positively charged when it recoils from radium A and 
that the “recoil-atoms” can be deflected by an electric field. 
Some experiments have lately been made to measure the 
deflexion suffered by the radium B when it passes through a 
strong magnetic field. For this purpose a powerful electro- 
magnet was constructed § capable of giving 10,000 lines per 
square centimetre over an area 9 cm. by 5 cm., with the 
poles 2 cm. apart. A glass tube of 2 cm. external diameter 


containing the apparatus shown in fig. 1 was placed in this 
gap between the poles; the glass vessel could be rapidly 


* Coffin, Proceedings American Academy, xxxix. No. 19, 1903. 

t+ Communicated by the Authors. 

t Russ and Makower, Phil. Mag. supra, p. 875. { 

§ We are indebted to Dr. R. Beattie for designing and supervising the 
construction of this magnet. 


Field of Radium B on Recoil from Radium A. 883 


evacuated when required. The wire W, of diameter 0°5 mm., 
coated with radium A by exposure to the emanation for 
ten minutes, was placed at the end of the aperture between 
the two metal plates A, B, 1 cm. long and 0°5 mm. apart. 
The “ recoil-stream” from W passed through this aperture and 
fell upon the plate C. The metal tube K served to prevent 
disturbances by stray electric charges on the glass of the 
containing vessel. In its path of 7:1 cms. from W to C, the 
recoil-stream was exposed to a uniform magnetic field. 

To carry out an experiment the active wire W was 
mounted in the position shown in fig. 1, the glass tube was 
evacuated as quickly as possible, the magnetic field applied 
and the recoil-stream from Wallowed to pass between the plates 
A and B and fall upon the metal strip C. The distribution 
of the activity on the plate C was subsequently measured in 
exactly the same manner as in the experiments on the electro- 
static deflexion by means of an «-ray electroscope. To 
obtain the magnitude of the deflexion suffered by the radium 
B while passing through the magnetic field, two experiments 
were performed, one as described and a second one with the 
field reversed. The distribution of activity over the plate in 
these two experiments is shown in Pl. XVIII. fig. 2, curves I. 
and IT. respectively. Now it had been shown by other expe- 
riments that the strip C and wire W could be removed and 
replaced very nearly in the same position, so that the distance 
between the positions of maximum activity in the two expe- 
riments just described, gives twice the deflexion suffered by the 
recoil-stream in each experiment. It will be seen from fig. 2 
that the distance between the two maxima is ‘645 em. The 
paths of the recoil-streams are circles and the positions of 
maximum activity Q and R on the strip C are due to matter 
projected from W describing circular paths passing through 
WPQ and WRP respectively. If PC=d, and WP=d,, 
then if QR=d and p is the radius of curvature of the path 
of the rays, we have that 


dp=d, (d,+d,). 
Since d;=6'1 cm. and d,=1 em., it follows that 
p=67°2 cm. 


Since for the experiments described the strength of the 


_ magnetic field was 10,800 gauss we have, with the ordinary 
notation, that 


a =Hp=7°'26 x 10°. 


884 Dr. Makower and Mr. Evans: Deflexion by Magnetic 


The method of finding Hp just described is, however, open 
to certain objections, for it is necessary to make two separate 
experiments with a direct and reversed magnetic field in order 
to obtain the deflexion suffered by the recoil-streams. A 
further disadvantage is that a somewhat large aperture of 
width, 0°5 mm., was used. Some other experiments were 
therefore made as follows with a narrower slit and finer wire. 
The apparatus used is shown in fig. 3. 


Fig. 3. 


An active wire W of diameter 0:3 mm. was mounted as 
shown in the figure, 1°25 cm. from the slit 8 which was 
3°6 cm. from the strip C placed to receive the recoil-atoms. 
The whole apparatus was enclosed as before in a glass 
vessel which could be quickly evacuated and placed between 
the poles of the electromagnet. The wire W was fixed in 
position, the field applied and the glass vessel quickly 
evacuated, and the radium B recoiling from the radium A 
on the wire allowed to fall on the plate for three minutes. 
The field was then reversed and the recoil allowed to 
proceed for another seven minutes until the radium A on 
the wire had decayed to an inappreciable quantity. The 
strip C was then removed and placed on a_ photographic 
plate in the dark. The radium B distributed over the strip 
would itself have little or no effect on the photographic plate, 
but asit decayed radium C was produced in situ, and this by 
reason of the rays given out by it made an impression on the 
plate which could be developed in the ordinary way. The 
result of this experiment is shown in fig. 4(Pl. XVIII.). The 
two bands on the plate are due to the radium B reaching the 
plate with the direct and reversed magnetic fields respectively. 
It will be noticed that the bands are of considerable width and 
their edges not sharp, indicating that the particles of radium 
B on reaching the metallic strip are scattered before being 
stopped. That this should be so was to be expected con- 
sidering the relatively low velocity with which the particle 
must travel. In spite of this scattering, however, it is 
possible to measure the distance between the middles of the 
two bands with reasonable accuracy by the method of pro- 
jecting an image of the photograph on to a screen by means 
of a lantern and measuring the magnification thus produced 


Field of Radium B on Recoil from Radium A. 885 


in a manner similar to that adopted by Rutherford in his 

experiment on the magnetic deflexion of the «-particles *. 
The distance between the bands was in this way found to 

be 2°86 mm. Since d;=3'6 and d,.=4°85 we have as above 


p61, 
and since the field was 10,700 gauss, we have 


Mv ~ 
é i 


This value is certainly more reliable than that obtained by 
the first method described above, and is in fair agreement 
with it. 

It is of interest to compare the value of Hp obtained with 
that to be theoretically expected. Now the momentum of 
the recoil-atom of radium B which is produced on the 
emission of an a-particle from radium A must be the same as 
the momentum of this a-particle, since an atom of radium A 
gives out only one «-particle when it is transformed into 
radium B. It therefore follows that if the charge carried by 
radium B on recoil is the same as that carried by an a- 
particle, the value of Hp for the radium B should he the 
same as for the «-particle from radium A. Now it has 
been shown by Rutherford + that the value of Hp of this 
a-particle is 3°48 x 10° or nearly half that for the recoil-atom 
of radium B. It therefore appears that the charge carried 
by the radium B is half that on an a-particle, or, in other 
words, the atom of radium B carries with it the same charge 
as the hydrogen ion in electrolysis; for it is known that the 
a-particle is associated with twice that charge tf. 

Although the experimental values so far obtained are not 
of very great accuracy, it is possible to calculate from the 
electric and magnetic deflexions of radium B the velocity of 


these particles and the value of =. Since the charge carried 


by the particles has been shown to be the sameas that carried 
by the hydrogen ion in electrolysis, a knowledge of the latter 
quantity gives the atomic weight of radium B. 

Taking the path of the particles in an electric field as 
approximately circular, from the experiments of Russ and 
Makower we have that with a field of 16,250 volts per 


* Rutherford, Phil. Mag. Aug. 1906. 
+ Rutherford, Phil. Mag. Aug. 1906. 
¢ Rutherford and Geiger, Proc. Roy. Soc. A., vol. lxxxi. 1908. 


886 Profs. Trowbridge and Wood on Groove-Form . 


centimetre the radius of curvature was 12°9 em. Thus, 
using electromagnetic units, 


2 
MV 
é 


and taking the result obtained from the photographic method 
of measuring the magnetic deflexion, we have 


Mv 


— =6'5.x 10%, 
e 
Hence v=3'23 x 10’ centimetres per second and < =49°7. 


0) Hg 
Now since for the hydrogen ion in electrolysis—=9°6 x 103 
mn 


the result of these experiments gives the value for the 
atomic weight of radium Bas 194. Considering the diffi- 
culty of the experiments this number is in good agreement 
with the theoretical value 214. Also it will be noticed 
that the velocity of the particles has very nearly the value 
3°27 x 10’ centimetres per second, calculated on the assump- 
tion that the momentum of the recoil-atoms of radium B is 
equal to that of the a-particle causing it to recoil, and that | 
the atomic weight of radium B is 214. 

It is with pleasure that we take this opportunity of 
thanking Professor Rutherford for the facilities he has 
afforded us for carrying out these experiments in his labora- 
tory and also for many valuable suggestions during the course 
of the work. 


XCVIII. Groove-Form and Energy Distribution of Diffrac- 
tion Gratings. By Aucustus TROWBRIDGE, Professor of 
Physics, Princeton University, and R. W. Woon, Pro- 
fessor of Experimental Physics, Johns Hopkins Unwversity*. 


eee ash no rigorous investigation has ever been 

made of the distribution of energy among the spectra 
of different orders formed by a diffraction grating, as a 
function of the wave-length of the light and the form of 
the grooves. The chief obstacle in the way of such a 
study is the difficulty of obtaining an exact knowledge of 
the nature of the furrow cut by a diamond point upon a 
surface of glass or speculum metal. Microscopical examina- 
tion teaches us very little or nothing in the case of such fine 
markings, and it is not safe to infer that the groove will 
conform at all to the ruling point. 


* Communicated by the hora 


,and Energy Distribntion of Diffraction-Gratings. 887 


It occurred to one of us that a promising method of attack 
would be to stamp or rule gratings with such wide grooves 
that their form could be determined with certainty, and then 
investigate the energy distribution among the spectra with 
very long heat-waves, 2. e. with “‘ residual rays” of various 
wave-lengths. 

The manufacture of these echelette gratings, and their 
behaviour with visible light, have been described in a pre- 
ceding paper *. 

The investigation was made with the large vacuum spectro- 
bolometer described in a previous paper by one of the 
present writers t. The gratings were mounted on the table 
of the instrument, and the slit illuminated with the radiant 
energy in question. Two groups of rays were used in the 
investigation, the residual rays from quartz with a mean wave- 
length of 8:6 w and the CO, radiation from a Bunsen flame, 
with a wave-length of 4°3 w or about half as great as that of 
the quartz rays. The smallest grating constant used was 
0:0123 mm., or seven times the width of the grooves on the 
gratings ruled on Rowland’s first machine, the largest, 05 mm. 
It is evident that when these gratings are used with the long 
heat-waves above referred to, the ratio of the grating constant 
to the wave-length is about the same as that which obtains 
in the case of visible light and the optical gratings in common 
use. The nature of the ruled surface of the echelette 
gratings used in the present investigation, and the method 
by which it was studied, have been described in the previous 

aper. 
: ih order to make a thoroughly satisfactory study of the 
distribution of the energy it would be advisable to keep the 
angle of incidence fixed (for example normal) and swing 
the bolometer or thermopile through the spectra. With the 
instrument at our disposal at the present time this was 
impossible, and it was necessary to make the spectra pass 
across the bolometer by rotating the grating. This compli- 
cates the discussion of the results in no small degree, for the 
energy distribution varies with the angle of incidence, as 
can be seen easily with an ordinary grating. We have, 
however, already obtained results which are in qualitative 
agreement with theory, and which show that the method is 
admirably adapted to the experimental investigation of the 
problem. We shall, in the present treatment, discuss the 
results by the Fresnel method, considering the interference 
between secondary wavelets originating on the surface of the 


* Supra, p. 770. Tt Supra, p. 768. 


888 Profs. Trowbridge and Wood on Groove-Form , 


wave-fronts reflected from the oblique edges of the grooves. 
As Lord Rayleigh has pointed out, this method holds only 
when the width of the groove considerably exceeds the wave- 
length of the light. 

In the present case, with our closest ruling, the groove- 
width was 1°5 times the wave-length of our longest waves, 
and it appears probable that in this case we are very near, 
if not beyond the point, at which we may safely employ the 
Fresnel treatment. 

In continuing the work it is our intention to employ 
waves of continually increasing wave-length, until the 
point is reached at which the spectra disappear entirely, 
which will give us the complete experimental solution of 
each case. 

In the case of the echelette grating the conditions are 
quite different from those which obtain in the case of the 
gratings usually considered, which act by opacity. For a 
wire grating, or a reflecting grating made by ruling black 
lines on a reflecting surface, the spectra of even order fall 
out when the widths of the operative and inoperative elements 
are equal. In the case of the echelette grating, practically 
the whole surface is operative, and if we place the eye, or 
better the objective of a microscope (focussed upon the 
grooves in the direction of a spectrum) we see a uniform 
blaze of light illuminating the entire surface. This means 
that the widths of the reflected elements of the wave-front 
are twice as wide as in the case of a grating of the opaque 
type having the same constant. 

Now in a grating of this type the spectra of even order 
disappear when a=), as a result of the circumstance that in 
the directions of these spectra each diffracted wave front is 
self destructive, 1. e. these directions are the directions of 
Fraunhofer’s minima of the first class, namely such as will 
make the path difference between the disturbances coming 
from the two edges of each reflecting element equal to the 
wave-length of light. In the case of the reflecting grating 
with its opaque strips, if we widen the reflecting strips and 
narrow the opaque ones, keeping the constant the same, the 
direction of the first class minima will move in towards the 
first order spectra, which will disappear when the. opaque 
strips become infinitely narrow. The same thing, however, 
holds for all the other spectra, for as we widen the reflecting 
strip the first class minima draw closer together, coinciding 
with the spectra of the second class (grating spectra) in the 
limiting case of opaque lines infinitely nirrow. If, however, 
we narrow the reflecting strips, keeping the grating space 


Energy Distribution of Diffraction-Gratings. 889 


constant, the first class minima move out and presently the 
spectra of the third and sixth orders disappear. 

Going back now to the echelette grating we find that in 
the ideal case, in which the reflected fronts build up an 
unbroken surface (7. ¢. with no inoperative or dark regions 
between them) we should expect all of the light in one 
spectrum, namely the one lying in the direction in which 
the reflected wave-fronts are travelling, the case being 
analogous to the reflecting grating with infinitely narrow 
opaque lines, except that in this case we find the light in 
a spectrum instead of in the central image. We must 
remember, however, that in this case we have chopped up 
the wave-front into linear strips, and that our reflected wave- 
front is built up of strips obtained from successive waves, as 
can be seen from fig. 1, in which we have the reflexion of a 


oy 1. 
\ Diretion of Lneident 
waves. 


V ea 
3 3 vs | 


train of four waves, numbered 1, 2, 3, and 4 from the 
echelette grooves. It is very questionable whether the 
upper wave-front 4, 3, 2,1, will behave as a plane-wave, 
2. e. travel out without diffraction, for each one of the 
elements of which it is composed has had to travel one or 
more wave-lengths before uniting with its neighbour. 

This is a question, however, which can be best answered 
by experiment. In the paper on the Hchelette grating the 
opinion was expressed that a concentration of light could not 
be obtained in a region narrower than that covered by the 
diffraction range from a single reflecting element*. Further 
consideration shows that this is not the case, for in the ideal 
case shown in fig. 1 the maxima of the first class coincide 
in position with the minima of the second class, and vice versa. 
In the case figured the reflected waves are travelling in the 
direction of the first order spectrum, and the path-difference 
between the successive elementary wave-fronts is 2X. 

* Supra, p. 777. 
Hil. Mag. 8: 6: Vol. 20: Nos 119. Nov FOLO, 3 N 


890 Profs. Trowbridge and Wood on Groove-Form and 


Neglecting the probable disturbances in phase continuity 
resulting from the breaking up of the wave into narrow 
strips, we should expect all of the energy in the first order 
spectrum. If, however, we work with waves twice as long, 
the path-difference will be * instead of X and we should 
find the energy about equally divided between the central 
image and the first order spectrum, which in this case will 
lie well to the left of the direction in which the reflected 
waves start. Grating No. 8, which will be described pre- 
sently, comes the nearest to fulfilling these conditions of any 
thus far examined. With waves 4°3 in length the first 
order spectrum lies nearly in the direction of the reflected 
waves, and contains 70 per cent. of the energy. With the 
“* Reststrahlen” from quartz (\=8°6) we have 34 per cent. in 
the first order spectrum and 66 per cent. in the central image. 
The preponderance in the central image is due to the fact 
that the “ oblique image ” (direction of reflexion) lies nearer 
to the central image than the first order spectrum for 8°6 p. 

A large number of gratings have been examined and the 
work is not yet completed. For a complete solution of each 
case, it is necessary to know whether any of the original flat 
surface has been left between the grooves. This is often the 
case with the coarser rulings, and results in the formation of 
strong central images when the gratings are examined with 
visible light. Hach grating element may thus consist of 
three strips, the two edges of the groove and the flat portion 
between. Thus far, but a single type of groove has been 
tried, viz. the one ruled by the 120° carborundum crystal. 
The angle at which the crystal was mounted with respect to 
the surface has, however, been varied over a wide range, as 
well as the depth of the groove, &e. Other types of grooves 
will be investigated with a view of finding the one best suited 
for work in the infra-red. It seems probable that « 90° 
groove will be the best, as with a groove of this type one 
edge can be made almost inoperative, and a larger proportion 
of the surface brought into play. A symmetrically placed 
90° groove with the light incident normally will be an 
interesting type to investigate, for in this case we have a 
two-fold reflexion in the groove, each element of the plane 
wave being broken into two, which are turned end to end 
and reunited, as can be seen by constructing the reflected 
rays for a surface of this nature. A 90° double mirror has 
the property of returning a twice-reflected ray back to its 
source, regardless of its direction, provided it cross the groove 
in a direction perpendicular to the groove. 


Energy Distribution of Diffraction Gratings. 891 


In fig. 2 let AB bea portion of the plane wave AD inci- 
dent upon the grooved surface. After the first reflexion it 


Fig, 2 
a yO NN RR RR er 
& B eos Inerd. Wave 


will occupy the position A’B’, and after the second A’ B”. 
The portion BC will be reversed in the same way, and the 
two portions will unite into the wave C,B,A,;. It seems, 
therefore, as if a surface of this nature would not interfere 
with the constancy of the phase along the wave-front, not- 
withstanding the fact that the wave has been chopped to 
pieces, and the pieces made to change places. This being 
the case, it appears as if we should have no diffraction spectra 
at all, in spite of the deep furrows. 

Just how a surface ruled with grooves of this type, with 
perfectly smooth reflecting sides meeting in a sharp edge, 
would behave is perhaps open to question. Whether a wave 
can be broken up into paired strips, reversed, and reunited 
into a plane wave without suffering diffraction, is a question 
which can probably be answered only by experiment. It 
seems possible that many of the anomalies exhibited by 
reflecting gratings can be explained by a two-fold or even 
multiple reflexion from the groove. It is doubtful, however, 
if multiple reflexions can be considered as taking place in a 
groove commensurablé in size with the wave-length. The 
investigation of gratings of this type will be taken up later. 
The present paper deals only with the behaviour of the 120° 
groove. 

We will now take up the individual behaviour of the 
gratings which have been investigated up to the present time. 

The arrangement of the apparatus was as follows :—The 
light from a Nernst filament, rendered parallel by a concave 
mirror, was reflected from three large polished surfaces of 
quartz and focussed upon the slit of the vacuum spectro- 
bolometer by a second concave mirror. The diffraction 


dN 2 


892 Profs. Trowbridge and Wood on Groove-Form and 


spectra were caused to pass across the bolometer strip by 
revolving the grating, which made the incidence angle vary, 
and necessitated the use of the formula for fixed telescope 
and collimator and rotating grating. Hach grating was 
studied with the quartz residual rays, and with the CO, 
radiations, and the curves representing intensity distribution 
plotted. ‘lhe areas of the curves were measured with a 
planimeter to determine the total energy in each spectrum. 
The “central image”? curve was always very much higher 
than any spectrum curve, but owing to its narrowness 
frequently contained much less energy ; in other words, we 
cannot take the deflexion at the central image, and in a 
given spectrum, as a measure of the energy distribution, 
since the radiation is not monochromatic. The curves were 
plotted on large sheets of coordinate paper, and cannot be 
reproduced very well, even on a greatly reduced scale. It 
has seemed best, therefore, to make a small chart, showing 
in a rather qualitative manner the positions and magnitudes 
of the spectra of different orders, obtained with each grating. 
Dotted lines represent the CQO, radiation (wave-length 4°3), 
solid lines the quartz rays (wave-length 8°6). To save space 
the central image curve, the height of which is sometimes 
70 or 80 times the width at the base, is shown on a much 
smaller scale. 
We will now take up the gratings individually. 


Grating No. 4. Constant, 0°0265. 


This grating was ruled on Rowland’s oldest machine fitted 
with a 15-tooth cam. The constant is therefore 15 times as 
large as that of the usual gratings. The reflecting planes 
made angles of 20°5 and 27°5 with the original surface. 
Examination with the microscope, by the red and green 
light method, described in the paper on the echelette grating, 
showed that both reflecting surfaces were good, and that the 
grooves were separated by strips of the original surface, 
which appeared black under the microscope, and were of 
such a width that the 27°5 edge (red) plus the black strip, 
was equal to the width of the 20°5 edge (green), a matter 
of importance in connexion with the disappearance of the 
second order spectrum. ‘The grating upon the whole is 
neither a very satisfactory nor an interesting one. It concen- 
trates light both 1o the right and left of the central image, 
giving the brilliant spectra of the first class alluded to in the 
previous paper, at angles of 40° and 55° for normal incidence. 
In each ease the important thing to determine is the position 
of the heat-ray spectra with respect to the blaze of light 


Energy Distribution of Diffraction Gratings. 893 


(1st class spectra) reflected by the oblique edges of the 
grooves. If one edge of the groove makes an angle of 20° 
with the original surface, and the light is incident “uormally i 
the blaze of light, or the oblique image as we will term it 
hereafter, is seen at an angle of 40°. In the case of the 
spectrometer readings, it will be found 20° from the central 
image, since, when the grating turns with the circle through 
a given angle, the ray turns through the double angle. The 
angular position of the oblique image on our chart (fig. 3) is 


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ge 36° 35° 20° js) viet 5° Cent Tmare 5° toe 153° 20 rig 
therefore given by the angle of the edge of the groove. It 
has been indicated by brackets. In the case of very coarse 
rulings it is merely an image of the source, but slightly 
diffused by diffraction ; with the finer rulings we have the 
broad maxima and minima of the first class (the positions of 
the superposed patterns due to the individual reflecting strips 
or the so-called oblique images, are indicated by brackets 
in fig. 3). 


ma 


| 


$94 Profs. Trowbridge and Wood on Groove-Form and 


In the case of grating No. 4, the heat-ray spectra were 
found between the oblique image and the true central image, 
as will be seen from the chart. The explanation of this is not 
clear and it seems probable that some grave mistake was 
made in its examination. 

It will be necessary to repeat the observations with this 
grating, for no spectra were found to the right of the central 
image, with the exception of a very weak one of the first 
order, with the CO, rays, and we should expect strong ones 
in the position of the oblique image at 27°5. 

The distribution of intensity was as follows :— 


Total area of central image and spectra... 100 


Quartz rays, first order, left ............... 36 
Central Umass 5... icc ct0\ eet 64 

CO, rays, first order, left ............... 32 
third ~.,. Sr fei eamcanee ea 12 

Oentral image. jiscsccctan ® 46 


The absence of the second order with the CO, rays probably 
results from the circumstance that the widths of the reflecting 
edges are equal to the inoperative surface between them. 
There are therefore gaps between the reflected wavelets 
as in the case of a wire grating for which a=L, and the 
spectrum of two orders higher or lower than the one in the 
direction in which the waves travel will fall out, 2. e. 
the waves are self-destructive in these directions. The 
oblique image lies in the direction of the 4th order spectrum 
for the CO, rays, consequently the 2nd order should be 
absent as was found to be the case. It is interesting to see 
that. with quartz rays we have a strong first order spectrum 
at this point. Our failure to find a strong 2nd order 
spectrum for quartz rays, and 4th order for CQ, rays, in 
the direction of the oblique image probably resulted from 
our failure to rotate the grating sufficiently to bring this 
region upon the bolometer. 


Grating No. 5. Constant, 0°0123 mm. 


Angle of reflecting planes, 22° and 30°. Microscopical 
examination showed that the 30° planes were very poor re- 
flectors ; they appeared covered with dark patches, showing 
that the surface was very ragged and there was no oblique 
image to the right of the central image, which was wholly 
absent with visible light. With the beat rays the intensity 
distribution was as follows :— 


| 
: 
: 
| 
. 
‘ 
‘ 
’ 
| 
: 
| 
1 
. 
1 


Energy Distribution of Diffraction Gratings. §95 


Quartz Rays. CO, Rays. 
First order, left ... 63 Second order, left 46 
S nt ees. Centralimage ... 54 
Central image ...... 30 


In this case the slit width was 0°5 mm., and the width of 
the bolometer strip was 0°5 mm., so that the resolving power 
was high. The deflexions of the galvanometer could be 
repeated with an error of less than 1 mm, (with total 
deflexions of 35 mm.). 

The first order spectrum for the quartz rays falls exactly 
in the direction of the oblique image, and contains 63 per cent. 
of the energy. With visible light there is scarcely a trace 
of any central image, practically all of the light going into 
the oblique image, yet with the heat-rays we find the central 
image quite strong. This is what we should expect, for 
the heat-rays are diffracted by the edges of the grooves to 
the same extent as are light-waves by an ordinary optical 
grating, and the concentration is not complete even when the 
direction of a spectrum coincides with that of the oblique 
image. 

With the CO, rays the first order spectra areabsent. This 
is in agreement with theory, for if we draw the reflected 
wave-fronts moving off at an angle of 44°, and make their 
width such as would obtain with a grating constant of 0123 
and angles such as specified, we find that if we reduce their 
width a trifle the path-difference between the disturbances 
coming from their edges, in the direction of the first order 
spectrum, is exactly 2. 

We should, however, expect the central image to very 
nearly disappear from the same circumstance, for in this case 
the path-ditference will be 2A. Its appearance is probably 
due to the circumstance that the ideal condition of fig. 1 is 
not fulfilled, z. e. there is a narrow dark region between the 
reflected wave-fronts. We have moreover disturbances from 
the other edges of the groove which have not been taken 
into account. 


Grating No. 6. Constant, ‘0123 mm. 
Angles of Reflecting Planes, 18° and 29°. 


The 18° edge was bad, showing little reflexion with visible 
light. The oblique image had a blue central maximum, 
bordered by red and yellow maxima. This peculiarity has 
been discussed in the previous paper, and was found to result 


896 Profs. Trowbridge and Wood on Groove-Form and 


from the circumstance that the 29° edge was double. The 
intensity distribution was as follows :— 


Quartz Rays. CO, Rays. 
Central image ...... 17 Central image...... 16'8 
First order, left ... 57 2nd order, left ...%a7 
al 3, MEME... 26 et 5). 
Ist. »,,:, cieht ee 


The oblique image lies at 29° to the left, midway between 
the 2nd and 3rd order spectra of the CO, rays, consequently 
they contain very nearly the same amount of energy. The 
first order to the left is absent, but it is present on the right- 
hand side, though feeble. 

The quartz rays show a remarkable concentration in the 
first order spectrum which contains 57 per cent. of the total 
energy. With the CO, rays we again find concentration 
owing to shorter wave-lengths, for the two spectra, which lie 
symmetrically to the right and left of the oblique image, 
contain together 76 per cent. of the energy. 


Grating No. 7. Constant, ‘0123 mm. 
Angle of Planes, 30° and 20°. 


The 20° planes were rough, reflected poorly, and gave no 
oblique image. The 30° planes were excellent, and there 
was no visible central image. The intensity distribution was 
as follows :— 


Quartz Rays. CO, Rays. 
Central image... 52 1st order, left ... 2 
First order ...... 48 > 1:0 oe 

2nd ;) op 4, Keke aaeeee 
Pat Be dates ae 
Central image ... 23 


Here again we have the first order spectra practically 
absent for the CO, rays, while the second order contains 
63 per cent. of the total energy, the largest thus far obtained. 
The third spectrum in this case is very weak. 

With grating No. 6 the 2nd and 3rd order spectra were 
of equal intensity, yet there is very little difference between 
the gratings when examined optically, except the very inferior 
reflecting power of the 20° planes in the case of No. 7. 
Owing to the powerful concentration of the energy in the 
2nd order spectrum, fainter outlying parts of the CO, band 
appear reaching nearly to the 3rd order spectrum of the 
strong portion. 


| 
, 
| 


Energy Distribution of Diffraction Gratings. 897 
Grating No. 8. Constant, ‘0123 mm. 
Angle of Planes, 8°. 


In this case the angle of the reflecting planes is so small 
that the maxima and minima of the oblique image meet the 
grating spectra to the left of the central image. These 
grating spectra and the central image are strong, and the 
colour distribution is most remarkable. The spectrum of 
the 3rd order, for example, may contain only green light, 
while another contains only red and blue, as shown by a 
coloured plate in the forthcoming edition of ‘Physical Optics’ 
(Wood’s). 


The intensity distribution with the heat-rays is as follows:— 


Quartz Rays. CO, Rays. 
Centralimage ... 66 Centralimage ... 22 
Ist order, left ... 34 Ist order, left ... 70 


Pea ELOTG con a 8 


In this case we find, for the CO, rays, the second order 
absent, and a very strong first order, which lies very near 
the point towards which the energy is thrown by the 8° 
reflecting planes. As in the previous case, the CO, band 
appears wide and distinctly resolved into a double band at 
the centre. The oblique image lies nearer to the central 
image than the first order spectrum for the quartz rays, 
consequently it receives the larger portion of the energy. 

This grating has been already discussed in connexion with 
the theory. 


Grating No. 9. Constant ‘0265 mm. 


The angle of the reflecting planes was very small, about 
6°, and most of the energy appears i the first order spectrum 
from the CO, rays, which were the only ones used in this 
case. 

The distribution of intensity was as follows :-— 


Kirst, order; lett} ).4%: - 40 
Second ,, i re ee 3 
vind. 4ssu1 sree 25ee3- 10 
Central image ......... 32 
First order, right ...... 9 


Here again we have concentration of energy in the first 
order spectrum as a result of the small angle of the reflecting 
planes. 


898 Profs. Trowbridge and Wood on Infra-Red 


Summary of Results. 


The results obtained thus far appear to be in excellent 
agreement with theory, and indicate that the method gives 
reliable experimental data regarding the distribution of energy 
as a function of the groove form. It indicates that diffrac- 
tion of the radiation from the reflecting planes prevents us 
from concentrating all of the energy in a single spectrum, 
but that with a properly sloped edge we can utilize as much 
as 70 per cent. of the energy. 

This paper is intended only asa preliminary communication, 
and the investigation of grooves of other forms will be taken 
up next. A more exact knowledve of the precise nature of 
the ruled surface is desired, and preliminary experiments 
have shown that it can be obtained by making sections with 
a microtome of celluloid casts of the surface. These com- 
bined with the microscopical examination with red and green 
light, and spectrometer determinations of the groove angles, 
will give us a very complete idea of the furrow. 

In the future it is planned to use more homogeneous 
radiation, by spectral decomposition of white light with a 
rock-salt apparatus, and study each grating with a wider 
range of wave-lengths. This will enable us to pass by 
gradual stages from the energy distribution obtained with 
the quartz-rays to that obtained with the CO, radiation. 


XCIX. Note on Infra-Red Investigations with the Echelette 
Grating. By Aucustus TROWBRIDGE, Professor of Physics, 
Princeton University, and R. W. Woon, Professor of Ea- 
perimental Physics, Johns Hopkins University”. 


1 the preceding paper we have reported a preliminary 

study of the distribution of intensity among the spectra 
of different orders furnished by the echelette grating. In 
the present note we propose to show that these gratings give 
us the highest resolving power that has yet been brought to 
bear upon the remote infra-red region of the spectrum. The 
form of the intensity curve of the “ Reststrahlen ” reflected 
from quartz, discovered by Rubens and Nichols, is shown 
in fig. 1,a, which is from a figure given by Coblentz. The 
longer wave-length maximum is considerably higher than 
the short wave-length one, and the minimum between is very 
shallow. A curve of practically the same form has been 
obtained in all other investigations. In fig. 1, } is shown the 
curve obtained with the echelette grating. The maxima are 


* Communicated by the Authors. 


Investiyations with the Echelette Grating. 899 


practically of the same height, and the minimum between 
them has an intensity of only about 1/3 that of the maxima‘ 


rs 


Fig. 1. 
teres e 8.41 px 
M, Kx 
A: 
M4 . 
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ee 
U ) 5 \ Ay) 
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“see 


in other words, it is very much deeper than when observed 
with a rock-salt prism. The slit width was only half a milli- 
metre, and the width of the bolometer strip was the same. 
This type of curve was obtained with all of the gratings 
The maximum at 13, figured by Coblentz, who worked with 
a single reflexion only, did not appear in our work, as it was 
lost by the threefold reflexion which we employed. The 
curve obtained with the radiations of CO from the Bunsen 
flame is shown in fig. 2. There is distinct resolution of the 
band into a double band in the case of every curve obtained. 
In addition to the strong band at 4:3 there are weaker 
maxima to the right and left. The curve figured was 
obtained with grating No. 8. 


$00 Infra-Red Investigations with the Echelette Grating. 


The wave-lengths of the maxima of the bands for the 
quartz rays and the radiations from the flame have been very 


Fig. 2. 


WY 


5.3) 4 
7 by a” 


' ares : a 


carefully determined. ‘the angle between the collimating 
mirrors of the vacuum spectrometer were determined, and 
the formula for fixed collimator and telescope and revolving 
erating. In the case of the quartz rays the following values 
were obtained :— 


8:42 8:90 
8°42 8°99 
8:40 8°80 
8°36 8:90 
8:49 8-90 
Mean ... 8:4lp 8:90pm 


For the flame radiations the following values were found 
for the three maxima, 4°2u, 4:4, 4:5u (these values were 
obtained from calculations made both from the second and 
third order spectra of grating No.6). With grating No. 5 
the values of the two brightest maxima were 4°32 and 
4°43. 


On Molecular Attraction. 901 


Previous work on CQ, is as follows. Julius finds bands at 
2°8 and 4:4. Rubens and Aschkinass find, in addition, a 
weak at 14:1. Water vapour has a band at 5-4. The smal! 
maximum which we find at 2°84 is undoubtedly a first order 
maximum on the concentration side which Julius located 
at 2°8. The two large maxima at 4°41 and 4°51 are first 
order spectra of the Julius band, found at 4:4, which he failed 
to resolve but which was distinctly resolved in the present 
case. The small band at 5°37 is probably first order for the 
water-vapour band which Rubens and Aschkinass found, 
while the other small band may be a first order band at 5°77 
(due to ?), or more probably a second order of the band 
at 2°84. If this beso, its wave-length in second order figures 
ought to be 2°89. 

No very great effort was made to get the highest resolution 
possible, and the results given are to be regarded rather as a 
by-product of the other investigation. 

It is interesting to compare the dispersion of the gratings 
with that of rock-salt and fluorite prisms :— 

In the interval 4y to 5m in the spectrum furnished by a 
rock-salt prism, the difference in angles of minimum deviation 
is 0° 15' of arc. This means 7''5 of arc change in the setting 
of the Wadsworth prism-mirror combination. Inthe grating 
spectrum grating of No. 8, 13’ of are corresponds to 0'1p, or 
1p would correspond to about 130’ of arc. Our dispersion 
therefore near the CO, band is nearly 17 times that of a 60° 
rock-salt prismin the same region. This region is about the 
worst part of the rock-salt spectrum, on account of the 
flatness of the dispersion curve at this point. The dispersion 
of a 60° fluorite prism is apparently 4 times that of a salt 
prism between 4u and 5yu; therefore our dispersion is about 
4 times that of fluorite in this region. Near the quartz bands 
with grating No. 5 we have about 150 of arc to ly, while with 
a 60° rock-salt prism there are 27’ of arc to lu. Here then 
we have 54 times the dispersion of a standard 60° salt prism. 

Further investigations with the gratings will be made in 
the near future. 


PELs 


C. On Molecular Attraction. ay 
To the Editors of the Philosophical Magaziney 
GENTLEMEN,— 

: oo a paper published in the October number of the 

Philosophical Magazine, Mr. Mills gives a résumé of 

some of his work on molecular attraction. May I be per- 

mitted to call attention to some points in connexion with the 
subject. 


j 
f j 


902 On Molecular Attraction. 


It can be strictly shown mathematically that it is impossible 
to determine completely the law of attraction between mole- 
cules from latent heat data, or in other words the law deduced 
should contain an arbitrary function of the distance of 
separation of the molecules and the temperature. It follows 
therefore that if we assume a certain law of attraction and 
deduce from it a formula for the latent heat and find that it 
fits the facts, it does not therefore follow that the law assumed 
is correct. In fact an intinite number of different laws can 
be obtained, each of which gives a formula for the latent heat 
agreeing with the facts. It is owing to this that different 
investigators have obtained different laws for the attractions 
between molecules. Attention was drawn to this point in a 
paper read before the Sheffield Meeting of the British Asso- 
ciation. A demonstration of the result stated cannot be 
given here, and I therefore beg to refer to a paper on the 
subject which will be published in this journal shortly. 

That we may deduce from latent heat data more laws than 
one, can be shown by an example. Assuming that the 
attraction between two molecules varies inversely as the 
square of their distance of separation, Mills obtained for 
the latent heat the formula A(p!* — pl/*), where A is a con- 
stant and p;, p> denote the densities‘of the liquid and vapour 
respectively. Now if we assume that the attraction varies 
inversely as the seventh power of the distance, we obtain the 
formula B(p{—p%) for the latent heat, where B is a constant, 
and this also agrees well with the facts (“‘ Equation of State,” 
Phil. Mag., supra, pp. 678 et seq.). We may therefore with 
equal justice assume that the attraction obeys the latter law. 

Further, it can be shown independently of the above con- 
siderations that the law of Mills cannot possibly account for 
the latent heat of evaporation, etc. According to Mills the 


attraction between two molecules is 3? where z is their 


distance of separation and K a constant. The heat of 
evaporation of a molecule may be taken as the energy 
expended in bringing it from the interior of a large mass of 
liquid to an infinite distance from the liquid; for which I 
have given a general formula, Phil. Mag. May 1910, p. 801. 


Substituting = for $(<)(= ¥. m,)? in the formula we obtain 
K 
L =a (e¥—pl®) 2-96, 
where m is the mass of a molecule and L the internal latent 


heat in ergs. In ithe case of ether at 273° this gives 
K=8-9x10-# dyne, K being the attraction between the 


On the Electricity of Mercury-falls. 903 


molecules unit distance apart. Now the gravitational 
attraction obeys the inverse square law, and its value for two 
ether molecules separated by unit distance should therefore 
be equal to the above value of K. The former quantity is equal 
to 1:84 x 10-*? dyne, which is much smaller than the latter ; 
and this law cannot therefore account for the latent heat of eva- 
poration. Apart from molecular attraction considerations the 
latent heat formula given by Mills is, however, of great interest. 
To be on safe ground the law deduced from latent heat 
data must contain an arbitrary function. But still the law 
may give some valuable information, for we might be able to 
prove that the arbitrary function cannot include some of the 
known parts of the law obtained, and these parts may bring 
out some important properties, and these must be true. 
Yours faithfully, 
Cambridge, Oct. 5, 1910. R. KLEEMAN. 


CI. Note on the Electricity of Mercury-falls and on very 


large Ions. 


To the Editors of the Philosophical Magazine. 

GENTLEMEN,— 
| gee eae to the paper by Mr. Lonsdale in your 

September number, on the “Ionization produced by 
the Splashing of Mercury,” I may perhaps mention to your 
readers, that Mr. Lonsdale’s results, which are certainly of 
great interest, are, however, not so new as may seem. Most 
of these results, including the large excess of the one kind 
of carriers (ions), the small velocity of these carriers, the 
influence of the surface of impact, are to be found in two 
elaborate papers by A. Becker, published in the Annalen der 
Physik, vol. xxix. p. 909, in 1909, and vol. xxxi. p. 98, in 1910, 
“Ueber Quecksilberfall elektricitét”’ (“On the Hlectricity 
of Mercury-falls”). Prof. Becker shows there also, that the 
smallest impurities of the mercury are of great influence,and he 
comes to employ very-carefully purified mercury. Moreover, 
in the same papers there are also to be found experiments with 
several amalgams and in other gases than air (H,, CO,), and 
the surface of impact is also further varied. 

As to the last poimt in Mr. Lonsdale’s summary of results, 
viz. the appearance of “neutral doublets,” reference may be 
made to the paper by K. Kahler (Ann. d. Phys. vol. xii. 
1903), who found quite the same appearance of new carriers 
of electricity in the air from waterfalls which had already 
passed an electric field and was therefore expected to be free 
trom carriers. According to the state of knowledge at that 
time, Mr. Kahler interpreted his result as ‘radioactive 


904 On the Statistical Theory of Radiation. 


property of the air from waterfalls.” Later on, we were not 
able to reproduce this unexpected reappearance of new carriers 
(cf. Aselmann, Ann. d. Phys. vol. xix. 1906), and this is quite 
in agreement with Mr. Lonsdale’s statement, that something 
not yet known seems necessary to produce this result. 

I may perhaps also mention here one of our more recent 
results bearing on the subject (P. Lenard and C. Ramsauer, 
Heidelberg Acad. of Sciences, 1910; also mentioned by 
C. Ramsauer at the Rad. Congress at Brussels), namely, that 
the smallest traces of water vapour, and perhaps also other 
vapours, are of great influence on the size of the carriers 
(ions) in gases—even at atmospheric pressure,—the carriers 
becoming very large when traces of such vapours are present. 
We have found this with carriers produced by ultra-violet 
light. The usual drying of the air by phosphoric acid is not 
sufficient to reduce the carriers to their smallest size; they 
became much smaller, if previous cooling to —70° C. was 
employed to purify the air. It may therefore be expected, 
that the large carriers from mercury-falls contain also many 
molecules of the liquid (mercury). 

Physical Laboratory of the Yours very sincerely, 

' University, Heidelberg, P. Lenarp. 
Sept. 26, 1910, 


CII. On the Statistical Theory of Radiation. 
To the Editors of the Philosophical Magazine. 

GENTLEMEN ,— ao ert, 6. 
| the Philosophical Magazine for August (p. 350) 
Sir J. Larmor gives reasons for supposing that my 
conclusion, that his modification of Planck’s theory of 
natural radiation does not evade the main difficulty that an 
atomic consutution of radiation must be implied, need not 
follow. 

Sir J. Larmor states that on his theory & need not be 
equal to the gas constant for one molecule but is indeterminate. 
To make this clear it ought to be shown that the calculations 
of k given by Boltzmann, Planck, and others can be modified 
so as to leave & indeterminate. This I think has not yet 
been done, and until it has been done the conclusion that & 
must have a definite value cannot be regarded as disposed of. 
The only reason why Planck was obliged to introduce the 
idea of finite elements of energy was that he found the only 


possible value of & to be that of the gas constant for one 


molecule. Yours very truly, 


Haroup A. WILSON. 


MAKOWER & EVANS. Phil. Mag. Ser. 6, Vol. 20, Pl. XVIII. 


rae 2: 


ACTIVITY 
” ~ o 
(eo) (e) (e) 


D 
O 


1654321012345 671 
DISTANCE IN MMS 


Fra. 4. 


THE 


LONDON, EDINBURGH, ano DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 
| DECEMBER 1910. 


CII. The Attraction Constant of a Molecule of a Substance 
and its Chemical Properties. By R. I. Kuneman, D.Sc., 
B.A., Mackinnon Student of the Royal Society *. ) 


|e the surface-tension of liquids and the latent heat of 

evaporation the writer t has deduced the nature of the 
law of attraction between molecules which gives rise to these 
properties of liquids. It follows from the nature of surface- 
tension and the heat of evaporation that the law obtained 
does not necessarily apply to distances between the attracting 
molecules less thau the distance of separation of molecules in 
the liquid state. Molecules and atoms may approach much 
nearer to one another than this distance, as happens for 
example in the polymerization of molecules and the com- 
bination of atoms to form a molecule. It is not impossible, 
therefore, that another force of attraction of a different 
nature exists besides that brought out by the above investi- 
gation, which operates effectively only when the distance of 
the attracting molecules or atoms is less than the distance of 
separation of molecules in the liquid state. Such a force of 
attraction, if it exists, would assist in producing chemical 
combination. It seems improbable, however, that any other 
force of attraction should exist than that which gives rise to 
surface-tension, and that this is therefore the force tending 
to produce chemical combination. Whether that is so or not. 

* Communicated by the Author; some of the results in this paper 
have been given in a paper read before the Meeting of the British Asso- 
ciation in Sheffield this year. 

+ Phil. Mag. May 1910, pp. 783-809. 
a Phil. Mag. S.'6. Vol, 207 No. 120, Dec: 1910. 30 


906 Dr. R. D. Kleeman on the AttractionConstant of a 


cannot be tested directly. There is some indirect evidence, 
however, supporting this supposition. Thus the attraction 
constant >,/m, denoting the sum of the square roots of the 
atomic weights of the atoms of a molecule in the law of 


attraction (=,/m,)? between molecules of the same kind, 


may be replaced by Xv, the sum of the maximum valencies 
of the atoms of a molecule. The quantity K in the ahove 
expression is the same for all substances at corresponding 
temperatures, and may therefore be a function of the ratio 
of the temperature of the molecules to the critical tempe- 
rature, and the ratio of z—the distance of separation of the 
molecules—to their distance of separation at the critical 
temperature. Its correct form was not indicated by the 
investigation mentioned: a later investigation * showed, 
however, that it must be principally a function of the 
temperature. 

If the force of attraction producing chemical combination 
is that given by the above law, we should expect that further 
relations of the quantity =\/m, of a substance with its 
chemical properties exist. The object of this paper is to 
point out some relations of this kind. These constitute 
further indirect evidence that the above law of attraction is 
the only one operating when chemical combination takes place. 

La a) 


The quantity en of substances, where T denotes a 


my 
given corresponding temperature, is of the greatest import- 
ance as its properties run parallel with the chemical properties 
of the substances. This parallelism appears in many ways. 


Thus the value of == for a substance and its substitution 
S/m, 


products is approximately a constant, but varies considerably 
from one set of substances to another. This is shown by 
‘Lables I. and II., using the critical temperatures ft of sub- 
stances, which by definition are corresponding temperatures. 


The tables contain also for comparison the values of Wes 

mm, 

for a number of other substances which are not substitution 
ees ¢; j 

products. The constancy of IG, is better in some sets of 

substances than in others. It is probable that a deviation 


iE ; lia 
at = 7 from constancy in a set of substances indicates a 
Ye/ my 


* Phil. Mag. Oct. 1910, p. 665. 3 
+ They were taken from Landolt and Bornstein’s Tables, 5th edition. 


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Molecule of a Substance and its Chemical Properties. 


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Molecule of a Substance and its Chemical Properties. 909 


departure from the normal chemical behaviour of the set. 
The grouping is however, very marked, the change in the 


value of SJ a as we pass from one group to another being 
1 
usually greater than the deviations from constancy of the 
values of one ern 
The values of i for SO, and CO, are approximately 


equal to one another ; ee behave as if both are substitution 
products of some primary compound. It may be noted that 
they are formed in a similar way, viz. by the burning of 
carbon or sulphur in oxygen. 
A number of expressions involving other quantities than 
the temperature can be deduced which possess the same 


property as Bee The expressions can be deduced by 


means of the relations 


Toe ee. oC) 
d= “(BY eS ei ucny inn de) 


mn 


Bee Vr a an 
L=—(2) (S/m,)?, WWE rod SPT EG) 
7/3 pe 
» = (2) 0 ECR oma alll C2 


given by the writer, where p, L, X denote the pressure of the 
saturated vapour, the latent heat, and the surface-tension of 
a liquid at the temperature T, and H, «", B, M are constants 
which have the same values for all liquids at corresponding 
temperatures, p denotes the density of the liquid and m its 
molecular weight. 
Let us write 
Y 
iat a Ue bea ame TM 


By means of equations (a) and (1) we obtain 


Bis ate ei, Y/m, 


3 
or HH”? m 1 


Pep y ea vm sat 


where v denotes the molecular volume of a molecule. 


(2) 


EEE —-s —_— 


ae 


910 Dr. R. D. Kleeman on the Attraction Constant of a 


Now at corresponding temperatures H is the same for all 


- liquids, and P is constant for a compound and its substitution 


products. The quantity Sm will therefore be constant 
1 


for a compound and its substitution products at corres- 
ponding temperatures. The ninth column of Table II. bears 
this out for the critical molecular volumes of a set of 
substances. 


By means of equations (2) and (b) we obtain 


preie )) X_ PRP« 
ie se (SV m,)"? or (Sa/m,)'? ae He Riabia t.- (3) 


Since — = constant for corresponding states, where EB 


denotes the potential energy of the surface film of a liquid, 


we have that (Sum. i2 is constant under the same 
conditions. ( my) 


From equations (c) and (2) we have 


* BePSV/ my hy = BYP : (4) 
Seog oe o/ii, OFS : 


where Lm=L, and is the molecular internal heat of evapora- 


L 


Fiery 
tion of a liquid. The quantity Vin is thus constant under 
1 


the same conditions as the above quantities. This is shown 
by the last column of Table IJ. The internal latent heats 
used relate to 2/3 of the critical temperature ; they have 
been calculated from the tables of latent heat given by 
Mills *. 

From equations (c) and (2) we have 


M?2P7/4 tea M2P7/4 
p= aie (vm), or vay me 


- 


Thus 371 18 constant under the same conditions as 
(20/m,) 


the quantities discussed above. This is borne out by the 
tenth column of Table II., using the critical pressures of the 


substances. Since the exponent in (29/m,)"" is i. the value 
* Journ. of Phys. Chem, vol. viii. p. 405 (1904). 


Molecule of a Substance and its Chemical Properties. 911 


of this expression will not vary much from one substance to 
another ; and the critical pressures will therefore not vary 
much in the case of a compound and its substitution 
products. 


We have then that the values of each of the simple 
v r E 


: T 
expressions Se (So/m,)*” (S/m)” (S/m)"” 


L, 
S/n. (eV my are constant for a compound and its 
AV Mm, 1 
substitution products at corresponding temperatures. 

Any function of these quantities will possess the same 
properties. It is of interest to note that the form of the 
function may be such that S4/m, does not appear in it. 


c 


Further chemical properties of the quantity Sian will 
1 


now be discussed. The chemical compounds considered will 
be taken in groups according to their chemical properties. 


Esters. 


iV 3 
The values of Dale for each set of isomers of the esters 
<a Mm 


are practically the same, as will be seen from Tables I. and 


satis 
II. In comparing the values of a oy therefore 


take the mean of the values of each set. These mean values 
are contained in Table III. (p. 912). 

The chemical formula for the esters may be written 
20+4(C+2H), where « denotes an integer. Now it was 


T. 
found that the value of S/m, MY be expressed by the 
equation m4 


; iY 
A,—{20+4(C+2H)}=(A,—20)—2a(C 4 2H) =. (ye 


where A, is a constant, and H, C, O have the same value for 


each compound. The values of (Ay—20) and (C+2H) were 
T, 
determined from the values of = 7= by the method of least 
L/m, 


squares, and found to be equal to 29 and 2°51 respectively. 
By means of these quantities the value of the left-hand side 
of the above equation was calculated for each ester, the result 


912 Dr. R. D. Kleeman on the Attraction Constant of a 


being given in the last column of Table III. It will be seen 


rity 


that the agreement with the values of Sein fairly good. 


TaBLe III. 
Romoeal 
Substance. Vie! ZV m,- | SV an’ 4220), 

M. formate, C,H,Og.........2004.. 487 18°92 25°79 24°98 

| Mean of type C,H,O, ........-..-) 5044 | 2438 | 2069 21°47 
4 9 Og, .necpenssese | 528:3 29 84 17°70 18°96 

"a fs) Oglh0O)) .pecevsps yy D47°5 35°30 15°51 16°45 

7" oy CaHH Og 2. censreees 5645 40 76 13°85 13°94 

Nye PR om PP 1 So Rn EE | 587°6 | 46-22 | 12°71 11°43 

Acids. 


The chemical formula for the acids contained in Table IV. 
may be written 20+a(C+2H), where a denotes an integer, 
and is thus the same as that of the esters. This expression 


Ay 
mav be connected with =—7— by an equation similar to 
) n S4/m, 7. q 


TaBLe IV. 
|_Te_ |4,—20) 
a0 /— ——e 2 
Substance. Te. AN iil) | ZV m, | —e(C+2H),) 

| Butyric acid, C,H,0O, ......... 611 29°84 20°47 20°14 
| Acetic acid,  ©,H,O, ......... 5945 | 1892 | 31-42 31:22 
Propionic acid, O38,0, .......-. 6129 | 2488 | 25-14 25°68 
; 


| Pee 


that obtained in the case of the esters ; that is, we may 
write 


A, —(20+a(C+ 2H))=(A,—20) —a(C+ 2H) = ay 


where A, is a constant. The values of (A,—20) and 
(C+2H) are, however, much larger than the values of the 
corresponding quantities of the esters. Thus it was found 


Molecule of a Substance and its Chemical Properties. 913 


by means of the method of least squares that 
(A,—20) = 42°3 and (C+2H) = 5°34, 
values which are about double the corresponding values for 


the esters. The value of the left-hand side of the above 
equation was calculated for each acid by means of these 


T, 
quantities, and a fair agreement with the values of Se 
obtained. shh 


Nitriles. 


The chemical formula for the nitriles contained in Table V. 


may be written N+aC+H, where « and 8 denote integers. 
Te 

It can be connected with the value of Sas in a similar 

way as before ; that is, we may write ce! 


te 
aia Y — a i) = ys = So 
A;—(N+eC+6H)=(A;—N) —eC—fH I 
The values of (A;—N), C, and H were calculated by the 


dt 
method of least squares from the values of =—— in the 
Y/m, 


table, giving 43°42, 2°051, and 1°339 respectively. Using 
these values, the values of the left-hand side of the above 
equation was calculated for the nitriles in the table. 


The agreement with the values of ven is fairly good. 


TABLE V. 
Te (A;—N 
= ays air ) 
Substance, Te. ZV m,. | Wm, —2C— GH. 

Acetone nitrile, C,H3N ......... 543°2 13°67 3774 35°30 
Benzonitrile, ©;H;N ......... 6992 | 32:97 | 21-20 22:37 
Butyronitrile, C,H;N ......... 5821 | 24:59 | 23°67 25°84 
Capronitrile, -@gHl;, Nos... G28) 351 17°52 16°39 

| Propionitrile, CsH;N ......... 558-1 | 1913 | 29-20 30°57 
| Toluylnitrile, O,H;N ......... 7238 3843 | 18-81 17°64 


The deviations from exact agreement that occur are greater 
than the experimental error that can occur in the determi- 
nation of the critical temperatures. The lack of the quantity 


914 Dr. R. D. Kleeman on the Attraction Constant of a 


T 
ST tec being exactly additive, probably indicates certain 


deviations in chemical properties of the substances. 
The values of H and C, it should be noticed, increase with 
increase of atomic weight. 


Ethers. 


The chemical equation for the ethers contained in Table VI. 
may be written aC+@H+0O, where a@ and B are integers. 
From what has gone before we should expect that there 
exists a relation of the form 

i 


A,—«C—BH—O= (A,—0)—aC—BH= 5. /na 


The values of (A,—Q), C, and H, determined from the 
four ethers, are 34°87, ~*277, and 1959 respectively. The 
values of the left-hand side of the latter equation calculated 
by means of these quantities are given in the fifth column of 


the table, and agree fairly well with the values of STm 
my 


The values obtained for H and C decrease algebraically with 
increase of atomic weight. The opposite occurs, we have 
seen, When H and C occur in the equation for the nitriles. 


TABLE VI. 
om | eee | 2. (a) 
Substance. | Te | > Vim. =m, ‘~aC—6H ~(0+2H). 
Allylethylether, C,H,,0..., 518 | 3130 | 1655 16:66 
Methyl ethyl ether, O,H,0 | 4407; 22:38) 1970 | 2003 20:07 
Methyl ether, ...... C,H,0 ...| 4026| 1692 | 2379 | 23-67 23-60 
Ethyl oxide, ...... ©,H,,0 M 465°6| 27:84 | 16-73 16-29 16 55 


The chemical equation for the three ethers at the end 
of the table may be written O+a(C+2H), and we have 


accordingly (As—0) —a(C4+ 2B) = where A; is a 


bf 


my 
constant. The values of (A;—O) and (C+2H) were found 
to be equal to 30°64 and 3°522 respectively. The values of 
the left-hand side of the above equation calculated by means 
of these quantities will be seen to agree well with the values of 


c 
SJ/m,' 


a , 
— 
‘ 


Molecule of a Substance and its Chemical Properties. 919 


The value of the expression (C+2H) obtained by substi- 
tuting for H and C their values is 3°64, which is approximately 
equal to 3°52 the value of (C+ 2H) just obtained in a different 
way. 


Amines. 


The chemical formula for the three primary amines in 
Table VII. may be written N+3H+a(C+ 2H) where @ is an 


Cc 


integer. We would therefore expect the value of 


SV im 
to be given by the equation 
ES 
A,—(N +3H+4(C+2H)) =(A,—N—3H) —a(C+ 2H) =F 
1 


where Ag is some constant. It was found in the same way 
as before that (Ag—N—3H)=41°27 and (C+2H)=7°01. 
By means of these values the left hand-side of the above 
equation was evaluated for each of the three primary amines 
in the table, giving a good agreement with the corresponding 


ialnes of, — =. 
SV my 
TaBuLE VII. 
— AN 
Substance, Te. | EN m,- ay — | (A, NoH)— ale 2Et 
2m, 
Ethylamine, CABIN ack 450 17°67 | 25°48 27°25 
Methylamine, CH;N ...... 428 12°21 35:05 34:26 
Propylamine, .CHN..;:..... 491 251d) | 2027 20°24 
Diethylamine, O,H,,N...... 489 25 99.|> LALO 18°57 
Dimethylamine, C,H,N ...... 436 17-67 | 2468 23°81 
Dipropylamine, C,HisN......| 550 39°51 13°92 13:33 
Triethylamine, C,H,,N...... 5382 39°51 13°47 13°47 
Trimethylamine, C3H,N ...... 433°5| 2313 | 23:06 23°06 
Pyridinj ue. CHgNie.c. 2) 6172} 26:05 | 23°69 25°09 


The chemical equation for the three secondary amines in 
| E/ my 
we would therefore expect to be given by an equation of the 


above form. The values of (A,—N—3H) and (C+2H) for 


the table is the same as the above, and the value of 


916 Dr. R. D. Kleeman on the Attraction Constant of a 


the secondary amines were found to be equal to 29°05 and 
2°62 respectively. These values, it will be noticed, differ 
considerably from those found for the primary amines. A 
fair agreement of the left-hand side of the above equation 


c 


with the values of “__ is also obtained in the case of the 
an my : 

secondary amines. . 

The chemical equation for the two tertiary amines in 


Table VII. is the same as for the primary amines, and is 


therefore connected with a by an equation similar to 
My 

the above. It was found that (A,—N—3H)=382°61 and 

(C+2H)=3:19. These values are very nearly equal to those 

obtained for the secondary amines. 

The chemical equation for the anilines in Table VIII. is 
N—5H+a4(C+ 2H), where a is an integer. The equation for 
Te we would therefore expect to be 
> Vm 
ne ; ‘ ‘ ie 
A,;—(N—5H+a(C + 2H))=(A,—N + 5H)—a(C+2H)= See 

The values of (A; —N+5H) and (C+2H) were calculated 

from the three anilines in the table by the method of least 

squares and found to be equal to 39°79 and 2°94 respec- 
tively, and these values then used to calculate the values of 


A . 
ee, PHO agreement between the values thus obtained 


SV my 


and those obtained directly is very good. 


TasLe VIII. 


_ (i 
Substance. Te. | San, sW= (A, —N-+5H)—a(C +2H). 
| | Wty 

ps Ee Na baat ut 

Aniline, ...... 10: 8. ee 6986 3151 | 2218 22-15 
Etby] aniline, CsH,,N ...... 6984] 4243 | 16-46 16:27 
Methyl aniline, C;H,N ...... 7016| 36:97 | 1898 19:21 

: 
. . . . . Te » @ . 
The organic alkali pyridin, whose value of SV mn is given 
m\ 


in Table VII., is usually classed with tne tertiary amines: © 


Molecule of a Substance and its Chemical Properties. 


SING 


Its chemical formula is the same as that of the anilines, and 


we may 


to pee also to pyridin. 


ee Ly 


S a 


therefore suppose the above equation for the anilines 
It is of interest that the value of 


pyridin is approximately given by the above 


equation, using the values of (A,—N+5H) and (C+2H) 
The value obtained in this way is 
25°09, which is approximately equal to 23°69, the value given 


found for the anilines. 


in Table VER 


Alcohols. 


The chemical formula for each of the two sets of alcohols 
in Table IX. is 0+2H+a(C+2H), where «& is an integer. 
We would therefore expect that 


As—(0+2H+a(C+2H))=(A,— 


O— 2H)—a(C+ 2H) = 


sie Dre Renee reo. 
Sm, my 


From the first set we obtain (A,—O—2H) =30°58, and 
(C+2H)=2°635, giving a fair agreement of the eloulared 


values of 


m, 


From the second set we obtain (A,— 


(C+ 2H) =2-29. 


found for the first set. 


lated values of 


with those obtained directly. 
O—2H)=30-08 and 


These values do not differ much from thuse 
The agreement between the calcu-. 


and those obtained directly is very 


The values of 


my, 
respectively 44:14 and 30° o3. 
with the second set, though belonging to it. 


good. ie 
TaBLE IX. 

Substance Re al ae pew (As—O-—2H)—a(C+2H) 

ance, ita 2 
faovanty? alcohol! *.i..263..-35- 5796 | 33°30 17°41 17°40 
Esobutyl alcohol | .5.77.2-2.5.-- 561:3| 27:84 | 2017 20°04 
Isopropyl! alcohol ...-...--...... 5076] 2238 | 22°69 22°68 
Amyl alcohol ~..:2ainee see 621 33°30 | 18°65 18°53 
Butyl-alealiols we. Stunner se 5601} 27°84 20°12 20-92 
Propyl alcohol .......-.-.. ----- | 536-7| 22:38 | 23-99 23-21 


for methyl and ethyl alcohol are 
These aleohols do not fit in 


Using the 


918 Dr. R. D. Kleeman on the Attraction Constant of a 


values of (Ag—O—2H) and (C+2H) found for the second 
set of alcohols in Table IX., we obtain for the calculated 


values of =——— respectively the numbers 27:95 and 25°31, 
my 
which we see differ considerably from those obtained directly. 


The reason for the disagreement appears to be that these two 
alcohols are polymerized «at ordinary temperatures, which 
gives rise to a displacement of the critical temperature. The 
writer has shown that the liquids which we know (from 
surface-tension and other considerations) to be polymerized, 
usually do not fit in with the critical constant relations that 
apply to normal liquids. The polymerization of a liquid at 
ordinary temperatures may thus affect the critical constants 
although there may be no polymerization in the critical 
state. 


Sulphur Compounds. 


The sulphur compounds in Table X. are too diverse in 
character to admit of being divided into groups the members 
of each of which are connected by a simple relation. The 
following relations may, however, be noticed. The chemical 
formula for the first six compounds is S+2H+a(C+2H), 
where a is an integer. We may therefore suppose that 
there exists the relation 
¢ oh . Te 
A,—(S+2H+a(C+2H))=(A, -S—2H)—a(C+2H)= S Jae 
Now, if we suppose that the value of (A,—S—2H) and 
(C+2H) is the same for ethyl methyl sulphide and ethyl 
sulphide we get, by subtracting one from the other, that 
(C+2H)=3'25. It we suppose the values of these quantities 
are the same for ethyl sulphide (one of the former compounds) 
and methyl sulpbide, we get (C+ 2H)=6°47. This value is 
almost exactly double the former. Again, if we suppose that 
these quantities have the same values for ethyl and isoamyl 
sulphydrate we get (C+2H)=32:23, which is almost exactly 
half the previous value. And if these quantities have the 
same value for ethyl and isoamyl sulphide (which corre- 
spond to the sulphydrates) we obtain (C+ 2H)=1°37, which 
is much smaller than the value obtained in the case of the 
sulphydrates. 

When the critical constants for a larger number of sulphur 
compounds are available some systematic relations should 
be possible of being discovered by the help of which thi 


compounds might be classified into sets. : 


Molecule of a Substance and its Chemical Properties. 919 


TABLE X. 
ihe Te 
Substance, Te. aS V my. = Want 
Ethyl methyl sulphide, C,HsS_ ......... 582'T 24-04 22°16 
Ethyl sulphide, (0) 05 AIS eerie 5077 29°50 18°91 
Methyl sulphide, (O)a18 [eh sae nanen 592 18°58 31°85 
Ethyl sulphydrate. OPE Sig nacre 22 501 18°58 26°96 
[soamyl sulphydrate, ©;H,,S ......... | 603-9 34:96 17°27 
Isoamyl] sulphide, CUSHEA Sif ssiuas: 664°2 62°26 10°67 
Allyl sulphide, Ch Shes: seat 6534 36°42 Nes 
Ethyl disulphide, O16 ioe baeneen 441-9 35°16 12:57 
Hydrocarbons. 


Table XI. contains the values of <= of a number of 
Vey 
paraffins and benzenes. The chemical formula may be 
written 2H+a(C+H). Let us suppose that 
pi 
A jel = C 2H ar — 
(Ajo )—a(C + 2H) San, 
and that the values of the quantities in this equation are 
the same for each paraffin. Subtracting each value of 


——— from the one preceding it we obtain the values 3°50, 


pa My 
2 00, 2°04, 1°43, 1:22, 92 for (\C+2H). They are not equal 
TABLE XI. 
Substance. | Te. -| SA m,. ie | Substance. d Wen > V tity aoe 
1y ; 


ed | 
a 
| 


Methane, CH, ...| 191-2} 7:46 | 25-63 || Heptane...C,H,, | 539°9| 40:32 | 13:39 


Ethane, C,H,...| 308 13-92 | 2213 | Octane eeCalty,) | 009) 4|.) 40°68 12-47 


Propane, C,Hg...) 370 18°38 20:13 || Benzene ...C,H, | 553°6| 26°76 20°68 
Ethyl 
Pentane, O;H,, | 4702) 29°23 | 16°04 benzene, CsH,, | 6194] 37°68 16°82 
Propyl 

Hexane, O©,H,, | 5078} 34°76 14°61 benzene, C,H,, | 5996) 43°14 13°90 


to one another, but decrease in value with the complexity of 
the molecules considered, showing that the quantities Ajo. U, 
H, have not the same value for each paraffin. This applies 


920 Attraction Constant of a Molecule of a Substance. 
also to the beuzenes in the table. It does not seem possible 
ry 


to draw any further conclusions from the values of =—*— in 
the table. ea | 
It will in general be found that if the molecules of a com- 
pound contain the same atoms as the molecules of another 
compound and in addition to these one or more atoms of any 
1 


. T 4 : . 
kind, the value of ——*— of the former compound is always 


SV m, 


smaller than that of the latter. 


Generul Remarks. 
The values of -—— or P of substances, where T denotes 
dy / my 
a corresponding temperature, may be expressed in terms of 
other quantities. Thus from equations (1), (2), (3), (4), and 
(5) we have 
T ! 4/3 ty 2 2/3 
a P=H() my, P= ys =e 
LV m, Me eh ai (24/m,) 
ba RMR lgtlg Pees ll 
RNS Re te oir, oa 7 ae 
B Xa / my M (S,/m,)"" a 
IE: fe 
At corresponding states each of the quantities (a S/m, 
, m 
Rage toad lien ig 
1/3 hee Vit 
(SV/m) (S/m,) (S/ m) 
a constant which is the same for each substance, and these 


G 


quantities therefore possess the same properties as ——, 
So/ my 


is equal to P multiplied by 


The theory of corresponding states is, however, only ap- 
proximately true, and the properties of these quantities may 


therefore not appear so marked as those of Te : 
‘ S/m, 


: r 
The properties of the quantity yeh or P of substances, 
My 
we have seen, usually run parallel with the chemical properties. 
One of these, stated in general terms, is, if ¢(H, C, O,...) 
denotes the general chemical formula for a group of sub- 
stances linked together by chemical properties, then the 
equation A—4(H, C, O,...) = ——-—.,, where A is a constant, 
aK 


applies to all the substances. ‘The various results obtained 
indicate the lines along which chemical compounds may be 
classified aceording to their physical properties. They would 
also be of use in aiding the classification from a purely 


Primary and Secondary y Rays. 991 


chemical point of view in pointing out the existence of 
ehemical relations. Thus if i¢ is found that certain relations 


T 
connect the values of —~— of certain chemical compounds, 
ain / My, 


we might expect that some corresponding chemical relations 
exist. The determination of the critical constants of chemical 
compounds, especially the critical temperature, becomes 
therefore of great importance. A more complete list of the 
critical constants than the one at present available may lead 
to the discovery of a number of other properties of the 


quantity Nya corresponding to certain chemical properties, 
A/ Ny 


besides those given in this paper. 


i 
Sm 


of a substance should be so closely connected with its 
chemical properties. We may state this quantity in a 
different way, but that hardly throws any light on the subject. 
The kinetic energy of a molecule is proportional to the 
temperature, and we may therefore define this quantity 
as the ratio of the kinetic energy of a molecule at the 
critical temperature to its chemical attraction at a given 
distance. 

Further relations of the quantity with physical and 
chemical quantities will be given in subsequent papers, A 
comparison of the various results obtained will probably lead 
to a definite explanation why the properties of this quantity 
run parallel with the chemical properties. 


Cambridge, July 20, 1910. 


There appears no obvious reason why the quantity 


CIV. Primary and Secondary y Rays. \ frond 


By D. C. H. Frorance, WA., MSe.* = \} 


] HEN y-rays strike a body it is well known that 
secondary radiation is produced. Part consists of a 
corpuscular radiation similar in character to §-rays; and 
part consists of a very penetrating radiation similar in cha- 
racter to the primary y-rays. Eve first proved that these 
penetrating secondary rays were of the y-ray type. Kleemant 


* Communicated by Prof. E. Rutherford, F.R.S. 
t+ Phil. Maz. Dec. 1904. 
{t Phil. Mao. May 1908. 


Beer hil. Mag. 8:6) Vols 20 No. 1202 Deer TNi0: ak 


922 Mr. D. C. H. Florance on 


examined closely the radiations from various metals, and from 
his results concluded that the primary and secondary y- rays 
could be divided into several homogeneous groups. Madsen* 
by a study of the ‘‘emergent”’ radiation, 7. e. the radiation 
emitted from a screen in the direction of the primary ¥ rays, 
divided the primary radiation into two homogeneous groups. 
One was “Shard” or very penetrating, and the other a soft 
group or one easily absorbed. He has shown that there 
exists a marked lack of symmetry in the quantity of secondary 
radiation emitted from the two sides of the plate, and in some 
cases a considerable difference in the penetrating power of 
the radiation. He considers that this secondary radiation is 
derived from the primary by a scattering process. 

The question of the distribution and character of the 
secondary y-rays is very complicated, and although a large 
amount of work has been done, many points still remain to 
be settled. It is of great importance, for example, to settle 
whether the secondary radiation of the y-ray type is merely 
part of the primary rays which have been scattered in their 
passage through matter, or is a true secondary radiation 
excited by the passage of the y-rays through matter. In the 
latter case, it is to “be expected that the secondary y-rays 
would differ in quality from the primary. One of the main 
difficulties of the subject is the apparent complexity of the 
primary y-rays, to which attention has been drawn by Eve, 
Kleeman, and Madsen. Soddy has found under special con- 
ditions that the y-rays from radium are absorbed according 
to an exponential law, and has concluded consequently that 
the radiations are homogeneous. This view is, however, 
difficult to reconcile with the evidence obtained by a study 

of the secondary y-rays f. 

The following experiments were undertaken to see if an 
conclusive evidence could be obtained to settle between these 
hypotheses. ‘The results given in this paper extend and 
somewhat modify those obtained by Madsen. The y-rays 
appear to be entirely heterogeneous, and the terms “ hard ”’ 
and “soft” can only be used for convenience and not to 
denote two distinct groups of homogeneous rays. 

The paper consists of two parts:— 

I. A short investigation of the initial absorption of y-rays 
by lead under ordinary experimental conditions. 


* Phil. Mag. March 1909. 
+ Note recent letters: Kleeman, Phil. Mag. July 1910; Soddy, Phil. 
Mag. August 1910. 


Primary and Secondary y Rays. 923 


II. Secondary y radiation. 
(1) The distribution of secondary y radiation : 
(a) produced by different thicknesses of the 
same material ; 
(b) produced by different materials. 
(2) The quality of the secondary y radiation. 
The variation of quality with 
(a) Position of electroscope. 
(6) Material of radiator. 
(c) Thickness of radiator. 
(d) Area of radiator. 
(e) Screening of radium. 
(3) A discussion of the question whether the secondary 
y-rays are true secondary rays or scattered 
primary rays. 


I. Inrtrat ABSORPTION oF y Rays. 


When y-rays are absorbed by a substance such as lead it 
has been observed by most experimenters that the coefficient 
of absorption decreases with an increase in thickness of the 
absorbing material. This has been generally explained by 
assuming the original radiation to be heterogeneous, and 
consequently the softer radiation to be cut out more in pro- 
portion than the harder radiation. That the y-rays from 
radium are heterogeneous is no doubt true, but the above 
assumption requires modification to explain all the cases that 
are likely to arise. Observers do not agree in the exact 
values to be assigned for the initial absorption of y-rays. A 
few experiments were performed to see what was the reason 
of this divergence. 


Apparatus. 


The electroscope was of lead, 3 mm. thick and 7 cm. cube. 
The top, bottom, and the two sides of the electroscope con- 
taining the windows were surrounded in lead about 1 cm. 
thick. The face through which the y-ravs penetrated was 
3 mm. thick. A screen of lead *7 em. could be used to test 
the quality of the radiation. The quality is determined by 
calculating X the absorption coefficient of this ‘7 cm. of lead 
from the relation I,=I)e~**, where I) is the leak in the 
electroscope before the lead screen is placed in position. 
The electroscope was supported on an iron pipe about 1} 
inches in diameter. The radium was placed near the edge 
of a table, and in some cases the lead screen was supported 


abi 2 


924 Mr. D. C. H. Florance on 


by string from a board overhead. Thus care was taken to 
prevent “secondary radiation from surrounding bodies. In 
these experiments the leak of the electroseope was fairly 
large, and no difference was observed when the small glass 
windows were screened by lead. 

Radium (20 mgrs. Ra) was placed 56 ems. from the electro- 
scope. The absorption coefficient was measured for narrow 
pieces of lead +172 em. thick placed against the radium. The 
length, breadth, and thickness were (1 x 3°5 x °172)em, 


Thickness of Screen, 


i | 
| 516 854 1:38 
| Thickness {| 33 | 172 2 cm. "G84.cm. 1:04 em. 1:55 em. 
| Yoram | -99 “0 65 | 58 


From this cap ceanatis it would seem that the lead screen 
simply cut out the less penetrating y- rays. 

Compare these results with “those obtained by other 
investigators. 


f Thickness | { ‘8 1:05 I's 18 
McClelland*... 2 | ~ 1 105em.] 13 cm. 1°8 cm. 2°3 cm. 

ea Ai eas 64 ‘D6 ‘48 “44 

' | “4 Ez 10 2-2 5:4 

Thickness | ‘ ; d 

Tuomikoskit... (= Pita tee ee 

‘ion Cee Ss 70 & 52 ‘50: 

| 1-21 1-79 2-4 

| Evet ti ab inc { 13 1-21 ‘al E 1-79 2:36 3-0 
Ratt di _ 

| Kies BT 56 | 46 46 


* Phil, Mag. July 1904. t Phys. Zeit. June 1909. + Pail. Mag. April 1906. 


As we have seen, Soddy has concluded that the true value 
for the absorption coefficient right from the initial stage 
should be A=°50. 

A glance at the above results shows discrepancies much 
too large to he ascribed to experimental errors. 

The radium still in the same position is surrounded with 
lead. A piece (11x5x1'4)cm. is placed directly in front 
of the radium and lead 1°5 cm. thick placed at the side. 


Primary and Secondary y Rays. 925 


The absorption coefficient is now determined for a screen of 
lead (11 x11x-°7) cm., when it is placed against the lead 
screen in front of the radium; and secondly, when it is 
placed against the electroscope. 


“7 cm. against the lead screen A="50 
ae tky Ms 5, electroscope A="55. 


This effect seemed to be caused by the secondary radiation 
emitted by the screen. 

A narrow screen 3°4 cm. wide and 1°3 em. thick is placed 
in front of the radium. A lead screen (13x 13x1'1)cm. 
is divided into three strips. One of these narrow strips is 
placed against the lead screen in front of the radium, and the 
coefficient of absorption is thus determined. 

The apparent absorption coefficient A="51. The two other 
strips are now added to continue the plate, and for this large 
plateX='46. The side portions of the screen which are added 
in this latter case do not cut off any of the direct primary radia- 
tion from the electroscope. They add, however, the secondary 
radiation due to the passage of the primary radiation. Hence 
the area of the absorbing screen will modify the value of the 
coefficient of absorption. LExperiments also show that bodies 
in the neighbourhood, from which a secondary radiation can 
be produced, will cause a variation in the value of results. 
The secondary radiation from the air due to the passage of 
the primary radiation would most probably be very small. 


Importance of secondary y-rays in the measurement of the 
absorption coefficient of primary y-rays. 


The radium is kept in a constant position 80 cm. from the 
electroscope. A large screen (20x 20x1:01)cm. can be 
placed in any position between the radium and electroscope. 
A lead screen ‘7 em. thick is placed against the electroscope 
to test the quality of the radiation. 


Position of large screen .../ Position I. Position IT. | Position ITI. 


Distance from electroscope...| 77:5 cm. 66 cm. lcm, 


No screen against electroscope; 5-Ud div./min. | 4°65 div./min. |4:29 div./min. 
Screen against electroscope...| 3°26 div.min. | 3:17 diy./min. |2°82 div./min. 


Absorption coefficient ......... A= "62 A="04 A="60 


It will be seen that the magnitude of the leak and also the 
absorption coefficient of the radiation change considerably 


996 Mr. D. C. H. Florance on 


with the position of the screen. In position I. the large 
screen does more than cut out the direct radiation--it adds 
the secondary radiation scattered from all parts of its volume. 
This quantity will decrease as the solid angle subtended by 
the screen at the radium decreases. On the other hand, as 
the screen approaches the electroscope it would be expected 
that the secondary radiation emitted would have an increased 
effect. Results point to the general conclusion that the pro- 
duction of the secondary radiation is the chief cause for the 
variation in the value of the absorption coefficient determined 
under different experimental conditions. The initial rapid 
change of the absorption coefficient is no doubt due to the 
rapid absorption of the soft portion of the primary radiation. 


In connexion with the secondary y-rays a few experiments 
were performed to test the quality and the amount of the 
primary y-rays passing through various materials. 

In this case a lead electroscope *6 em. thick was used. 
The thickness of absorbing screen was ‘624 cm. of lead, 
which was placed against the electroscope. The 20 mgrs. of 
radium was surrounded by -208 em. of lead. (See fig. 1.) 


Fig. 1. 


LLECTROSCOPE 


E is the electroscope in the position of direct radiation. 

E, and E, are positions of the electroscope for the measure- 
ment of secondary radiation. 

The radium cannot be placed symmetrically as regards the 
radiator as it is required to have E, as near the position E as 
possible without intercepting the direct radiation. 


Piimary and Secondary y Rays. 927 


The different radiators are supported by string. 
No radiator R= FTE 
5 ems. of carbon A==°68 
10 ems. of carbon 7~A="65 © 
Zrems. Of rol” * N= "6 
Seems! oniren’’ ***A="d 
-416 em. of lead N= "025 


The area of the radiator is constant (11x11)cem. Each 
radiator has had, therefore, a hardening effect. It will be 
shown later that the relative values of these absorption coefii- 
cients remain the same for each position of the electroscope. 

Consider now the amount which passes through these 
radiators. In fig. 2 are plotted the curves showing the 


seale-div./min. Fic: 2. 


JON/ISATION. 


SCALE FOR ABSCISSE 
/cr. FB 


THICKNESS X DENSITY 
Group A shows absorption of the direct y rays through different 

thicknesses of three radiators. Side of electroscope is 6 mm. thick. 
‘troup B, side of electroscope is 12 mm. thick. 


aA = 


92: 


os 


Mr. D. C. H. Florance on 


relation between thickness of radiator x the density, and the 
amount of radiation passing through as measured by the 
electroscope in scale-divisions per minute. It will be noticed 
that the curve for carbon falls between those for iron and 
Jead. This is unexpected, but it may be due to the fact that 
the radiation has to pass through the side of an electroscope 
6 mm. thick, which would cut out the radiations emitted 
from different radiators in a varying degree. The second 
group of curves shows the effect of placing 6°24 mm. of lead 
against the electroscope. The initial drop is not so great in 
this case. For equal weights per unit area the amount of 
radiation passing through varies for different substances. 

A comparison in the following table is made of the quan- 
tities passing through different radiators which give the same 
coefficient of absorption when examined by lead. 


| Mass per unit area | | 


Radiator. | Thickness. | in grs. ‘Quantity. r. 

Giebon mr cm, ran 22 mr aan 
| AOR | xesces 2 cm. 14°77 17 6) 

Lead ...... "25 em. 28 26 65 


A greater amount of the primary radiation passes through 
the lead than through the carbon, yet the absorption co- 
efficient is the same in both cases. There is no difficulty in 
explaining this if we assume that the y-rays of radium are 
heterogeneous. For the sake of clearness consider the y-rays 
divided into a hard and a soft group. Probably the soft 
group is in excess of the hard group, The results of expe- 
riments in the second part of this paper show that there 
is more scattering in a substance like carbon than in lead. 
The relative amount of scattering and of absorption of the 
two groups will most likely vary according to the material 
of the radiator. It is well known that lead cuts out a 
soft radiation much more rapidly than does a similar weight 
of iron. Therefore, when the primary radiation strikes the 
lead radiator, the softer portion will be cut out much more 
in proportion to the harder than in the case of the carbon 
radiator. Ifa radiator produces a scattering of the primary 
radiation, then the sorting out process will be simply a 
difference in degree for the two groups by different radiators. 
There is no need to suppose there has been a change in type 
of the primary radiation. 


Primary and Secondary y Rays. 929 
II. Srconpary y Rays. 
(1) Lhe Distribution of Secondary y Radiation. 


Apparatus.—The electroscope was of lead 3 mm. thick and 
7 cm. cube. In the first experiments, it was supported on a 
wooden arm which could be revolved so that the electroscope 
moved round the are of a circle of radius 25cm. ‘The centre 
of this circle was approximately the centre of the radiator. 
Lhe radium, about 300 mgrs.* of RaBr,, was contained in a 
platinum vessel, and the electroscope was screened from the 
direct radiation by a mass of lead. The arrangement was 
similar to that shown in fig.1. The radiator consisted of iron 
plates 11-1 cm. square. The electroscope was turned into 
the different positions, and the readings taken with and 


div./iain. Fig. 3. 


FOsITION OF 
LLECTROSCOPE 


AMOUNT OF SECONDARY FRADIATION. 


5 ms. WU cnes. \ 
THICKNESS 


Relation between amount of secondary radiation and thickness of 
iron radiator. 
without the radiator. Seven positions of the electroscope 
are taken; the first one measures the direct radiation and the 
last one the secondary radiation at right angles to this. The 
other positions are intermediate. 
In fig. 3 curves are plotted showing the effect of varying 


* This was kindly lent for the purpose by Professor Rutherford. It was 
sealed up in order to determine the rate of production of helium from it. 


930 Mr. D. C. H. Florance on 


the thickness of radiator from 1:05 em. to 11 em. for each 
position of the electroscope round a quadrant of a circle. 

Eve has shown (Phil. Mag. Dec. 1904) that for an increase 
in thickness of radiator, the emergent secondary y radiation 
increases rapidly till it reaches a maximum, and then it 
decreases. The curve thus obta‘ned can be expressed mathe- 
matically by the difference of two exponentials. Madsen 
(Phil. Mag. March 1909) has shown the same effect. The 
‘‘ineident” secondary y radiation, 2. e. the radiation turned 
back in its path, has also been shown to be represented by 
an expression K(1—e-2"), where K is a constant, A, the 
coefficient of absorption of this secondary radiation, and d 
the thickness of the plate. Experimenters in attacking this 
problem have kept their ionization vessel in one position 
and have made it large to obtain the greatest secondary 
effect. With 300 mers. of RaBr, there was sufficient secon- 
dary radiation to allow measurements to be made with a 
small electroscope and to be carried out round the are of a 
circle. Hence it was found that instead of a sharp line of 
demarcation between the emergent and incident secondar 
y radiation, the one gradually changed into the other. Curves 
illustrating this would change in form from that represented 
by (e7*¢—e-**) to (lL—e7 424), 

Considering the complexity of the radiation and the im- 
perfections of experimental arrangements, it is not to be 
expected that there would be any simple mathematical relation 
between the quantities measured; and the equations proposed 
by former experimenters are certainly inadequate. 


Relation between the amount of secondary radiation and position 
of electroscope for certain thicknesses of radiator (tig. 4). 


In these curves for each thickness of radiator the amount 
of secondary radiation is plotted radially. By continuing 
the curves an approximation can be obtained of the amount 
of the secondary radiation which passes through in the direc- 
tion of the primary radiation, and also of the amouni of 
“incident” secondary radiation. A similar approximation 
can be obtained from fig. 3. 

It is important to notice that a considerable portion of the 
total y radiation striking the radiator is converted into 
secondary. For instance, the leak in the electroscope when 
there is no radiator is 215 div./min. The leak when there 
is a radiator of iron 2:1 em. thick is 103 div./min. When 
the electroscope is in a position just outside the direct radia- 
tion the leak is 11 div./min. With the present arrangement 
it is difficult to determine the total quantity of secondary 


Primary and Secondary y Rays. 951 


radiation even approximately; but by integrating over the 
distributed curves there appears to be about 20 to 30 per cent. 
of the ionization due to the secondary radiation emitted from 
the radiator. 


Position of electroscope. 


Curves showing relation between position of electroscope and the amount 
of secondary radiation when the thickness is constant for each curve. 


A study of the curves shows that as the thickness of 
radiator increases the point A will approach 0 more rapidly 
than the points on the radial lines. It is not desirable at 
present to attempt to attach much meaning to the exact form 
of these curves, as it is possible that the shape would vary 
with the arrangement of the apparatus. The radium could 
not be placed symmetrically with respect to the radiator, 
but had to be placed against the lead screen. The gradual 
change in volume of the radiator due to a change in thick- 
ness from 1:05 to 11 cm. undoubtedly has a disturbing 
influence. 


It was thought possible that these results might be affected 
by secondary radiation from surrounding bodies, quite apart 
from the radiator itself. To test this point and to examine 


932 Mr. D. C. H. Florance on 


the quantity of radiation produced by different radiators the 
apparatus was set up afresh. A new electroscope was made 
of similar dimensions to the original one, but the thickness 
of the sides was 6 mm. A lead screen was also made for 
the small glass windows and a lead cap for the ebonite 
support of the leaf system. The electroscope was supported 
on an iron pipe so that it could be turned round an are of a 
circle. The radium (20 mgrs.) was surrounded by 2°08 mm. 
of lead. The radiators were supported by string from a 
beam overhead, so that secondary radiation from surrounding 
bodies was reduced toa minimum. ‘The arrangement is the 
same as shown in fig. 1. 

The direct radiation w.s first measured through the various 
radiators. The results were plotted with the ionizations in 
the electroscope as ordinates and the weight per unit area 
as abscisse. These curves (fig. 2) have already been 
referred to. 

To examine the secondary radiation two definite positions 
were taken and the results plotted in figs. 5 and 6. 


SECONDARY fADIATION. 


THICKNESS X DENSITY 


Electroscope turned through angle 25°. Comparison of amount of 
secondary radiation for different thicknesses of different materials. 


: 


Primary and Secondary y Rays. 933 
Fig. 6. 


AMOUNT OF SECONDARY FADIATION 


THICKNESS X DENSITY 


Electroscope at an angle 55°. Comparison of amount of secondary 
radiation for different thicknesses of different materials. 


Position I. The angle between the normal position and 
this position is approximately 25°. The carbon 
radiator produces more secondary radiation than does 
iron or lead. 

Position LI. The angle is approximately 55°. In this 
case the curve for carbon has fallen below that for 
iron, and even after 12°5 cm. of carbon the maximum 
amount of secondary radiation had not been reached. 
Lead reaches its maximum value and then decreases. 


A quantitative relation between the amounts of secondary 
radiation emitted by different radiators was looked for, but 

no evidence of such a relation has been discovered. The 
absorbing action of the 6 mm. side of the Elpcirescape would 
tend to mask any relation. It seemed probable, too, that 
variations in the amount of secondary radiation would be 
produced according to the experimental arrangement. In 
the present case, the volume of a certain weight of lead 
differed greatly from a similar weight of carbon. 

This is borne out by the result of a special experiment. 
For example, five sheets of lead were spaced out over a dis- 
tance of 8cm. They were held together by four thin brass 
rods so that they could be easily suspended in position, and 


934 Mr. D. C. H. Florance on 


thus they corresponded to the condition in which the large 
blocks of carbon were used. These readings were compar od 
with those obtained with five similar sheets of lead tied 
together. For the direct radiation, the rate of leak was 
2 per cent. greater in the case of the lead fastened closely 
together. But as the electroscope was turned round the 
secondary radiation from the radiator occupying the large 
volume produced a leak as much as 20 per cent. greater than 
that produced from the lead when tied together. This is 
what might have been expected, as the secondary radiation 
instead of getting absorbed has in the one case a chance of 
escaping, and this is more marked the further the electro- 
scope is moved from the direct line of radiation. In the 
present arrangement, where the radius of the are is about 
22 cm., the large volume of a radiator such as carbon may 
not give results comparable with those obtained for 1 cm. 
thick of lead. 


(2) The Quality of the Secondary y Radiation. 

The apparatus, as previously explained, was set up so as to 
reduce to a minimum the secondary radiation except that 
due to the radiator itself. A Jead screen 3 mm. thick was 
placed over the window so as to avoid any constant radiation 
that might get through them. The electroscope was in such 
a position, therefore, that when the radiator was placed in 
position none of the original radiation entering the electro- 
scope was stopped; but there was simply an increased leak 
in the electroscope due to the secondary radiation produced 
by the radiator. ° 

The quality or the penetrating power of the radiation was 
measured by placing a screen 6:24 mm. of lead against the 
side of the electroscope, which was 6 mm. thick. “Readings 
were taken without the radiator, first without the lead screen, 
secondly with the lead screen; then similar readings with 
the radiator. A large number of readings were taken, and 
the mean value is given. In these experiments radium 
emanation was generally used as a source of y-rays and a 
suitable correction was made for its decay. 

The first table shows clearly that for ail radiators the 
secondary y radiation gradually becomes softer as the electro- 
scope is moved further away from the normal position; and 
that an increase in thickness of radiation hardens both the 
primary and the secondary. The ratio of the absorption 
coefficients for any two metals or for different thicknesses of 
one metal keeps constant for each position of the electroscope. 

For example, the quality of the direct radiation passing 


Primary and Secondary y Rays. 


935 


Relation between coefficients of absorption of the secondary 
y radiation. 


The area of each radiator is (11:1 x 11'1)cm. 


Radiator. 


5 em. carbon . 
10 em. carbon . 
2°2.cm. iron ... 
5 cm. iron 
"416 cm. lead . 


Direct radiation. 


The coefficient of | 


absorption is 


00 


Electroscope at 
angle 25°. 


The coeffivient of | 


absorption is 
=1-20 
== JIBS) 
el 
=1:05 
= ba I | 


Electroscope at 
angle 65°. 


The coefficient of 
absorption is 
= ie 


=1-70 
=1-68 
=1-55 
=1-65 


Lifect of area of radiator. 


Radiator. Area. Hlectroscope at angle 80°. 
Fem. orion +... (Lieu T)y ent A=1°82 
2'2. cm. of iron ...| (22X22) cm. A=1°96 


through 2 cm. of iron is the same as that passing through 

10 em. of carbon. For any position of the electroscope this 

equality of ratio seems to hold true. This points to the con- 

clusion that the secondary radiation is the primary radiation 

scattered. If the radiation was a true secondary radiation 

it would be expected that the quality would depend on the 
material. 

In the second table it is shown that an increase in area of 
the radiator causes the secondary radiation to become softer. 
This is no doubt due to the fact that as the area increases, 
a more oblique secondary radiation will come from the 
radiator. 


Lffect of Sereens round Radium. 


Experiments were made to examine the effect of different 
screens round the radium. ‘The electroscope was 3 mm. 
thick, and 8°75 mm. of lead was used as an absorbing screen. 


Radiator. | Thickness. Radium unscreened. ls Screened. 
ae AR a: Be te ——— 
Carbom 2.cccnk 10 cm A =2°68 N= Lop 
head tee ae: 2-5 em A=1 60 


936 Mr. D. C. H. Florance on 


Two narrow blocks of lead, each 1°5 em. thick, were used 
as sereens to the 300 mers. of RaBro. The position of the 
electroscope was at right angles to the normal. 

From the table it is seen that the screen has a hardening 
effect. Similarly it was shown that any other screen always 
hada hardening effect. The ratio of the absorption coefficients 
remains the same for each radiator, and the results go to 
show that this ratio keeps constant for each position of the 
electroscope. 

Lead has always been used as the absorbing screen in these 
experiments, as the changes in coefticient of absorption are | 
much more marked than with any of the lighter substances. 
Yet similar results are given for screens of iron or zine. It 
is well known that lead will cut out the soft radiation to a 
much greater extent than iron. This holds for the secondary 
y-rays as well as for the primary. 


j 
; 


| Radiator. (‘875 cm. of Pb. | 1°05 cm, of Fe. 


| -875 em. of Pb. | 1-05 em. of Fe. 


| App= "70 AFe ='28 App =2'68 Ape ='56 


10 em. of C ... 


25m. of Pb...| App="46 pe 2t || App=160 | Ape="49 


| Direct Radiation. 


Klectroscope turned through 90°. 
H 


The absorption coefficient is first determined for *875 cm. 
of lead, then for 1:05 em. of iron. Hence, while the absorp- 
tion coefficient of the radiation changes from *70 to 2°68 
when measured by *875 cm. of lead, it only changes from 
"28 to °56 in the case of iron. Carbon shows this etfect to a 
less extent than iron, 

All metals will send outa radiation of the same quality 
provided the right thickness of radiator is used. 


Incident secondary y-rays. 


A few experiments were carried ont on the radiation 
emitted from the surface of the plate against which the 
primary rays strike. This radiation from iron and lead was 
softer than the emergent secondary radiation. This suggests 
that the softest radiation is most scattered. This incident 
radiation is similar in type to the emergent secondary and to 
the primary radiation. 


Primary and Secondary y Rays. 937 
4 y fray 


Summary. 


(1) Secondary y-rays are emitted from both sides of a 
plate exposed to y-rays. The “incident” secondary is in all 
eases softer than the “‘emergent” secondary. There is, 
moreover, a gradual change from the quality of the primary 
to that of the secondary emergent, and then to that of the 
secondary incident. The quality therefore depends on the 
position of the electroscope. 

(2) An increase in area of the radiator softens the secondary 
radiation, z. e. the quality depends on area of radiator. 

(3) An increase in thickness of the radiator produces a 
hardening of the primary and of the secondary emergent 
radiation. The quality depends on thickness of radiator. 

(4) For radiators of different material the quality varies. 
But if the right thickness for each radiator is chosen, then 
the quality of the primary and secondary radiation is inde- 
pendent of the material of radiator. With any two radiators 
the ratio of the absorption coefficients keeps approximately 
constant for any position of the electroscope. 

(5) The effect of screening the radium is to harden the 
secondary. The screen seems to harden the secondary radia- 
tion from carbon in the same proportion as it hardens the 
secondary from lead. This hardening is also proportional to 
the hardening of the primary as measured by the absorption 
coefficient. 

(6) The secondary radiation is heterogeneous, and this 
supports the view that the primary radiation is heterogeneous. 

(7) There is a gradual decrease in the quantity of secondary 
y radiation from that which emerges from the radiator in 
the direction of the original radiation to that which is returned 
in the reverse direction. 

(8) The curves showing the relation between quantity of 
secondary radiation and thickness of radiator change gradually 
in form for each successive position of the electroscope round 
the arc of a circle. 

(9) The lighter materials produce more secondary y radia- 
tion than the heavier materials. A greater weight, however, 
is required of the lighter materials before the maximum 
amount is reached. 


Discussion of the Results. 

In the foregoing results there is nothing to suggest that 
the secondary y radiation is a true secondary excited in the 
material of the radiator by a transformation of the primary 
rays. In such a case it would be expected that each element 


Piul. Mag. S. 6. Vol. 20. No. 120. Dec. 1910. a Q 


7 


—e eee 


938 Dr. J. W. Nicholson on the Approximate Calculation 


would give out a characteristic radiation. Experiments show 
that with proper conditions every substance can be so chosen 
as to give a similar type of radiation. It is important to 
notice that Bragg and Madsen (Phil. Mag. Oct. 1908) have 
shown that the character of the B radiation caused by y-rays 
is independent of the atom in which it arises, and depends 
solely on the nature of the y-rays to which it is due. The 
present investigation shows that this is also true for the 
secondary radiation. 

The quality of the secondary y radiation shows no sudden 
change from that of the primary. There is simply a gradual 
softening the more the secondary radiation is deflected from 
its original direction. The gradual softening is the same for 
every radiator. Other inv esti gators have shown that p- “rays 
are scattered in their passage throu gh matter. The scattering 
of y rays appears to be analogous to the scattering of 6-rays. 
The primary y-rays possess a wide range of penetrating 
power. The softening of the secondary radiation that has 
been observed is the result of this heterogeneity of the 
primary rays. ‘The softer radiation is more scattered than 
the harder radiation; as the radiator is increased in thickness 
more of the harder gets turned aside, and in consequence 
we get both the hardening of the primary and of the 
sec ondary, The hardening is due in the one case to the 
cutting out of the softer “radiation, and in the case of the 
secondary to the addition of a more penetrating scattered 
radiation. There is mo evidence of selective absorption, 
The production of this secondary y radiation is undoubtedly 

a scattering effect, as Madsen had concluded from previous 
exper iments. 


I desire to thank Professor Rutherford for the use of large 
quantities of radium and of radium emanation, and alse for 
his suggestions in the course of this work. 

Physical Laboratories, Manchester. 


CV. The Approximate Calculation of Bessel Functions of 
Imaginary Argument. By J. W. Nicaouson, M.A., D.Sc.* 


tH the British Association Report for 1908, some formulze 

were given suitable for the rapid tabulation of Bessel 
functions whose argument is purely imaginary and large, 
and whose order may be of any magnitude. The same results 
apply if the order is large, and the argument of any magni- 
tude. A proof was not appended, and the object of the 


* Communicated by the Author. 


of Bessel Functions of Imaginary Argument. 939 


present note is to supply a short proof. The corresponding 
formule: for functions of real argument have been very 
completely dealt with in a series of papers in the Philosophical 
Magazine*. The asymptotic expansions of functions of 
imaginary argument present only one type instead of the 
three in the case of real argument, and their treatment can 
therefore be given briefly. It is most conveniently deduced 
as a special case of that of the general associated Legendre 
functions P;'(w) and Q;(), which has been developed in a 
recent paper Tf. 

The functions of order m and argument iz satisfy the 


equation 
2 2, 
qe te (1+ Sy =0, eh oa 


da® xdax ¢ 


where « is itself real, and they are usually defined in the 
forms . 


am T ies 
[= (2) SS Pu (vt) = <TD |: cosh (2 COs ¢) sin?” p dd 


i x? ot 
= spesp it eateeit eae tt SO 
and 


‘ a\m V4 : ‘ 
K, (@)= G) rep \ dpsinn ide “Pee. eae Fy. 
the latter function vanishing exponentially when z is large. 
Let P?(u), Qr(u) be the general associated Legendre 
functions of argument pu, degree n, and order m. A compre- 
hensive definition of these functions for all values of these 
three quantities has been given by Hobson}. They are the 
functions which, when m and 7 are positive integers, may be 
expressed in relation to the ordinary zonal harmonics P,(y), 


Q,(“#) by the equations 
Pa(u) = Ait pig 2. 
Qi (u) = (#1). d* [du . Qa) 


when y is greater than unity, the only case needed for our 
purpose. But in the proof contained in this paper, restriction 
of the order and degree to integer values is not necessary, 
and the final results derived for the Bessel functions are true 
for any real value of m. 

% Dec. 1907; Aug. 1908; July 1909; Feb. 1910. 


+ Quarterly Journal, April 1910. 
{ Phil. Trans. 1896 A. p. 443 ef seq. 


3Q2 


940 Dr. J. W. Nicholson on the Approximate Calculation 


With these definitions, a well-known formula due to Heine 
shows that 


| (2) = Lt n—"Pn' (cosh), . ae 


and a companion formula may be readily derived as follows :— 
When w is greater than ‘unity , and m+4, n—m+l1 are 
positive, Hobson * has shown that 


m en" a(n+m) ow 1a 
Qn (4) = Ge ae << ) th Le \ 


sinh?” w dw 
(u + 1+ /p?- —1.cosh w)2t™*1 


(6) 


where a(s) is Gauss’ function, identical with rete or if 
s be an integer, with s! 

Write w=cosh x/n, where n tends towards infinity. Then 
(u?— 1)!" tends to the value (v]n)™, and a(n+m)/o(n—m) 
to the value n?”, 

Thus 

nar mee ~ ’ 2m 
Lt n-™Q"(u) = ae wont al ws sinh?” w dw 
— mF Hy ite Lt (145 cosh w ) 


nr=D 


\ntm+1 


But 


n+m+l1l 
Lt 1 zie ~ cosh w) _ e-7 cosh w, 


nr—=ow 


and therefore we deduce by the definition (3), 


ed @) Se Ue Br a (cosh =) 2 oe 


uo 


which is the required companion formula to (5). 


Asymptotic expansions. 


It is now possible to derive the asymptotic expansions of 
the Bessel functions IL,,(a) and K,,(v) from those of the 
Legendre functions. The latter will be quoted from the 
writer’s paper +, for the case of argument greater than unity. 


Writing 
PP) — emer (YM) ama) ={ 2G) A a Thet 


7 COS NT ToH(n ie (w?—1) 


SQ) <om) ee 


a(n—m)(42—1) 
c L. ce. ante, 


+ Quarterly Bea April 1910, pp. 250-252. 


of Bessel Functions of Imaginary Argument. 941 
Thenit f= mn, 7 = pw —1, 
T= 07 /2n+1)i(P+)-2+A3( +h) -3 +05 (7 +h?) -F+...} 


rm (vem BC) Fal) 
ays [ite marr Tg Bay 


ee Bilt Nude +e—T+h 
‘ log +/+ P— 15 — yolog EEE 


+$hlogs— |, uty sgl) 


where & is less than unity, and the coeflicients of types X,, 
Hr are given by 


= 1, As = —g(AP’— 1) /(n +3)? — 
4°4 (n+3)?—27}\3= — 64? (2 — 3h?) + 3 (4k? — 1) (284? —9)/8f (n +3)?—1}. 
and in general 
k*(k? —1)(r—2)(r—4)(r—6)Ar—6 + h?(2 — 3k”) (7 —2) (r —3) (7-4) ry 
+ (r+2){h? 4+ 3k2(r—2)?—(r—2)?2 br,» 
+ (r—1){4(n 44)" 1a, = 0, . (10) 
whereas the p’s are defined by the identical relation 
L+pmyot+poo?+... = (L+A go +A5074-...) 7). (11) 


We proceed to the limit when n is infinite and m finite, 
so that k=0. In this case, 


1 
POV yr 
L (2h 
4 fy eee pee | ERE EUS 2 
Los. ia(s 12m ), 


and soon. In fact, these limits are the coefficients which in 
the notation of previous papers dealing with the Bessel 
functions of real argument, were denoted by 


—Az, Nas — Ne, aa 


with m taking the place of n. Similarly, in the formula 
for t, n74,, n*yo,... must be replaced in the limit by 


2 a NS 
where the y’s are now the coefficients of earlier papers *. 


* Vide e.g. Phil. Mag. Feb. 1910, p. 240. 


—————— Le hl. lL UL hl OC hl! 


942 Calculation of Bessel Functions of Imaginary Argument. 


In terms of the old notation, therefore, on reduction, the 
limiting values become 


E d A d \ \ 
 — {1-242 305 2% (753) = = ie x 


x4 Vx? +m 7>—im log — Vat tm? a 
V2 +m? —m 
T = wf (2?+m’)-t—A, (a? +m?)-3+...b 2 2 (12) 
where the coefficients are defined by 


1 1 
Ao —- 3? — 37 (27—96m?), 
Ne == = (4640m?— 1125 —640m*), 


A(s+3)rA.43+ (s+ 2)®Aszi + 2m?s «(s+ 1) (s+2)rAs4 
+m's.(s?—4)As_3 = 0, . (13) 
and the identity 


1 pg + pg? + oo. = (14+ Agetryo?+...)-%. . (14) 


The limiting forms of the substitutions (8) become 


+ 
Lt Bard 3 (cosh “)= Lt t (=) et, 


it n—Meumn () (cosh= ~) = Lit cE 9 


where Q”” has now been rejected in the first substitution, as 
proportional by the second to e~* which is very small, for only 
moderate values of «?+m?, in comparison with é. 


Finally, therefore, by the use of (5) and (7), we obtain 
the results 
a deh z : 
T(z) = (a) € 


ei! mo 
Kote) = a 


where 
T = x} (2+ m?)-2—Do(e? + m2)—3 + Qa(a?2 +m?) -2— .,.f; 


lg ppt Wester a De 
hs Pemdm M4 i as 

(2? tent, 
(oF mei 


and the coefficients are given by (13, 14). 


x | (e2+m®)}—Jm log (16) 


On Non-Newtonian Mechanical Systems &c. 943 


These are the formule given in the British Association 
Report. 

So far as tabulation will ordinarily be required, it will be 
sufficient in general, even for only moderate values of w or m 
(2>10 or m>10) to take the first terms of T and ¢ only, if a 
three-figure accuracy is required. The order of accuracy 
possessed by the formule is similar to that of the ordinary 
semiconvergent expression for Jo(v) where @ is real. 

The first approximations may be written 


Ea) = (Qrx cosh Bree! B—B sinh 8) 
er ai me dacleosm 2 )r ce ost e-Bay” 

where @ is defined by m = wsinh 8. 
A usefal substitution in the final formule has been sug- 


gested to me by Prof. Alfred Lodge. If an angle @ be 
chosen such that 


(17) 


== im bane’, 
then 
i= m(see§ +loojtant6); js... .d8) 


and this logarithm has already been exhaustively tabulated. 
Thus the tabulation of the Bessel functions may be performed 
very rapidly, and this applies also when the higher approxi- 
mations are used. 


= SSS 7 — 
r 


CVI. On Non-Newtonian Mechanical ‘Systems, and Planck's 
Theory of Radiation. By J. H. Jeans, M.A. PRS.* 


Ji LANCK’S treatment of the radiation problem, 
introducing as it does the conception of an in- 
divisible atom of energy, and consequent discontinuity of 
motion, has led to the consideration of types of physical 
processes which were until recently unthought of, and are to 
many still unthinkable. The theory put forward by Planck 
would probably become acceptable to many if it could be 
stated physically in terms of continuous motion, or mathe- 
matically in terms of differential equations. Larmor f has 
recently made an extremely interesting suggestion as to how 
it might perhaps be possible to do this, but has not so far 
carried out the analysis necessary to determine whether his 
suggestion leads to a soiution of the difficulty or not. 
The question discussed in the present paper includes that 
* Communicated by the Author. 


+ Bakerian Lecture, 1909, Proc. Roy. Soc. A. vol. Ixxxiii, and Phil. 
Mag. xx. p. 390. . 


944 Prof. J. H. Jeans on Non-Newtonian Mechanical 


raised by Larmor’s suggestion and is in brief as follows :— 
Can any system of physical laws expressible in terms of con- 
tinuons motion (or of mathematical laws expressible in terms 
of differential equations) be constructed such that a system 
of matter and esther tends to a final state in which Planck’s 
law is obeyed ? It will be found that the answer obtained is 
in the negative. 


General Dynamical Investigation. 


2. We shall assume a law of causation—namely, that the 
state of the system at any instant is determined by its state 
at the previous instant, and that this state can be specified by 
the values of certain definite quantities 91, Po, .++ Pa, which 
we shall call the co-ordinates of the system. We shall first 
examine the consequences of assuming that time is con- 
tinuous and that these co-ordinates vary continuously with 
the time. 

3. If we construct an n-dimensional space, a single point 
in this arias namely the point whose co-ordinates are 
Dis Po +++ Pny Will represent the state of the system at any 
instant. A knowledge of the dynamical or kinematical laws 
obeyed by the system would lead directly to a knowledge of 
the paths or trajectories traced out in this space by the 
representative points as they follow the different possible 
motions of the system. We must not, in the present investi- 
gation, assume any special dynamical laws, but the general 
law of causation enables us to suppose that through every 
point in the generalized space there is one and only one 
trajectory, and that as a point moves along a trajectory, and 
so follows the motion of a system, its velocity at any point 
depends only on the co-ordinates of the point and not on the 
time. 

In the usual manner, we imagine every region of the 
generalized space which represents a physically possible state 
of the system to be filled with so many representative points 
that the whole collection of points may be regarded as 
forming a continuous fluid. The law of causation now states 
that this fluid moves along fixed stream-lines and that the 
velocity at any point remains constant. 

The initial distribution of density of the imaginary fluid in 
the generalized space remains entirely at our disposal. Since 
the motion is along fixed stream-lines with velocities fixed at 
each point, this initial distribution of density can be so chosen 
that, as the motion progresses, the density at every point of 
the space shall remain always ‘equal to the initial density at 
the point. We elect to arrange the initial distribution of 


Systems, and Planck’s Theory of Radiation. 945 


fluid in this way, and the motion of the fluid becomes “ steady- 
motion”’ in the usual hydrodynamical sense. ‘The state of 
the fluid is the same at all instants of time, so that we need 
only discuss it at one single instant. 

The mass of fluid, considered at any single instant, may be 
compressed, distorted and dilated, in such a way as to become 
of uniform density at every point*. After this distortion 
a hydrodynamical steady motion taking place along the 
distorted stream-lines will represent all possible motions of 
the dynamical system under discussion. Let us take new 
orthogonal Cartesian co-ordinates in this new (distorted) 
space, to be denoted by P,, Ps, ... Pu. 

4. The hydrodynamical condition for steady motion is 
ORs is 
Obst) 
so that we have seen that corresponding to any system of 
Jaws of motion of the dynamical system, at least one set of 
co-ordinates can always be chosen such that equation (1) is 
satisfied identically. And if there is one such set of co- 
ordinates there must necessarily be an infinite number, for 
a homogeneous fluid can be strained in an infinite number 
of ways so as to remain homogeneous. 

For example, if the motion of the dynamical system is 
governed by Newtonian laws, one set of co-ordinates which 
satisfy relation (1) is found in the Lagrangian co-ordinates 
and momenta, while other sets are obtained by taking 
Pi, Po, ...P, to be any series of linear functions of the 
Lagrangian co-ordinates and momenta such as determine a 
set of orthogonal lines in the generalized space. 

5. The mass of fluid moving in the generalized space pro- 
vides a basis for the introduction of the calculus of proba- 
bilities. 

At this stage it may perhaps be permissible to draw 
attention to a point which is often overlooked in _ the 
application of this calculus to problems of statistical mechanics, 
namely that any discussion of probabilities is meaningless 
until the basis of calculation of the probability is clearly 
stated. The question “ What is the probability that the 
entropy of a gas shall be W ?” is, unless a definite basis of 
probability is stated, as meaningless as the question “ What 
is the probability that the temperature of a gas shall be T, or 
that the gas shall be hydrogen?” Also, for the application 


be 


PE SEOUL HO a Fy 


* This is obviously true for a 1, 2 or 8-dimensional space and a proof 
by induction is easily constructed to extend to 2-dimensions. 


946 Prof. J. H. Jeans on Non-Newtonian Mechanical 


of the calculus to be legitimate we are not compelled to choose 
any one particular basis for the calculation of probabilities. 
We may select any basis we please, and the use of the calculus 
of probabilities will be legitimate provided we retain the same 
basis throughout the w hole inv estigation. 

In the present investigation we shall agree to say that the 
probabilities of a system being in states A or B are in the 
ratio of W, to Wz if the regions of the generalized space 
occupied by points representing systems in states A or B 
are in the ratio of W,4 to Wz. Or, in simpler language, in 
estimating probabilities, we think of the system as being 
selected at random from all systems in the generalized space, 
equal volumes of the space having equal chances of selection. 
This way of estim: iting probability leads at once, as we shall 
see, to Boltzmann’s relation between entr opy and probability. 

Let the points representing systems in different states 
A, B, C... oceupy regions which are in the ratios 
Wa: Wz: Wo.... Then, if a system is selected at random, 
the probabilities of its possessing characteristics A, B, C . 
are in the ratios Wa: Wg: Wo.... From the steadinene of 
the hydrodynamical motion, it also follows that if the system 
is selected at random and allowed to follow its natural motion 
for any time ¢, the probabilities of its possessing characteris- 
tics A, B, C ... at the end of this time will be in the ratios 
Wa: Wp: We.... And if the system is not initially 
selected at random, but starts from a known state, and moves 
for an indefinite time under its laws of motion, the probability 
of its possessing characteristics A, B, C ... at the end of this 
time will in general also be in the ratio Wy: Wz: Wo.... 
But this requires obvious modifications if the system is so 
started that at the end of infinite time it must inevitably have 
characteristics X, Y,Z.... The statement is then only true 
if Wa : Wz: Wo... measures the ratio of those parts of the 
space in which the characteristics AX YZ, BX YZ, CX YZ, ... 
obtain. 

Let A, B,C... now be characteristics of different parts 
of the system, such that the co-ordinates involved in the 
specification of any one characteristic are not involved in any 
of the others. Then the whole system may possess two or 
more of the characteristics simultaneously, and the probability 
that it possesses them all is of the form 


W=KW,i We Wo).':d) eee 


where K isa constant. The value of W is obtained by pure 


multiplication of Wa, Ws, Wc ... because the co-ordinates 
are orthogonal ; it is in no way necessary to suppose that: 


Systems, and Planck’s Theory of Radiation. 947 
A, B, U... are independent events. We put S=& log W, 


and $ is then Boltzmann’s measure of the entropy, proba- 
bilities now being measured on the basis provided by the 
generalized space. 

6. Let E,, E,,...be the energies of those parts of the 
system with which the properties A, B, C... are associated, 
and let E be the total energy given by 


H=E,+E.+ ee ° ° ° e . (3) 
The total entropy 8 is given by 
S=klog Wathklog Wet ...t+klogK. . . . (4) 


The characteristics A, B, 0... may be chosen so as to 
determine the partition of energy. To be precise, let 
characteristic A be satisfied if E, hes between H,’—4e, and 
E,/+4e, ; let B be satisfied if E, lies between H,'—4e, and 
E,/+4e, and so on. Let it be assumed, as a property 
of the system, that if left to itself, it will assume a state 
in which the energy is divided in a definite manner, namely 
one in which H,, E,,... become equal to E,’,: H,’..., 
at least to within small ranges e;, €,... Then W must be 
equal to unity for these values of E,, E, ..., and this is not 
only the maximum value of W, but is greater than the sum 
of all other values. It follows that S also must bea maximum, 
when H,, H,,... have the values E,’, E,’, ... subject to 
condition (3). The analytical condition for this is, in the 
usual way, that H,’, H,’, ... shall be given by the system of 
equations 


dS _ 38 _ 
ace a 


combined with equation (3). 

We can find the value of each fraction by supposing that 
part of the system is a perfect gas. We may assume this 
part of the system to obey the Newtonian laws, so that 
its co-ordinates P,, Ps,...P,, may be supposed identical 
with its Lagrangian co-ordinates and momenta, and its 


energy H, will be of the form 
Die os eee ° e ° ° ° e ° (6) 


the sum extending to m terms. The value of W, is now 
proportional to the volume of the region of the generalized 
space in which S,P,? lies between E,—3e, and HE, +4e, 
and is therefore of the form cE®”~‘e,, where c is a constant. 


948 Prof. J. H. Jeans on Non-Newtonian Mechanical 


Hence 
a8 2 ui km 
36, = * 9m, 28 Wa= 5, Gm—D= > + @) 


since m may be supposed very great. 

If T is the absolute temperature of the gas, and R the gas 
constant, the value of KE, is 4mRT, so that the value of 
OS/OH, becomes k/RT. 

If & is taken to be identical with R, then equation (5) 
becomes 

os _ Aas 1 
Seon + 


giving the second law of thermodynamics. 

7. This method of procedure shows the second law of 
thermodynamics to be more general than any system of 
dynamical laws ; the same can at once be shown to be true 
of the theorem of equipartition of energy. For suppose that 
any other part of the energy, say E,, can be expressed in the 
form given by equation (6), the summation now extending to 
nterms. The value of We, can be calculated in the same 
way as W,, and, just as in equation (7), we have 

0S — Rn 

ok, 2H, 
since k is pow identical with R. Since, by equation (8), 
0S8/oOE, must be equal to 1/T, it follows that 

E,=4nRT, 


expressing the law of equipartition of energy. The same 
result is clearly true if E, is any quadratic function of n of 
the co-ordinates. Moreover, if E, is a linear, cubic, bi- 
quadratic or any other homogeneous function of the co- 
ordinates, the result is still true in a modified sense, provided 
that E, is necessarily always positive. Tor if 


H.=f,(P3, Py, 6 wits P.), 


a homogeneous function of degree s, then W, is the volume 
of the generalized space included between the surfaces 


f(Pa Lael atid y Po j= Se. 


This is of the form cE,”*~'€, where cis a constant, so that 


aloe 1) _ mk 
= 5 = E, sHo 


me 


Systems, and Planck's Theory of Radiation. 949 


Hence we have E,=mRT/s. Thus in any part of the 
energy which is expressible as a homogeneous and necessarily 
positive function of the co-ordinates, the average energy of 
any m co-ordinates is proportional to m and to T ; but this is 
exactly the theorem of equipartition of energy. 

Hixperimental knowledge of wave-motion seems to place it 
beyond question that the energies of waves of different 
frequencies must be represented by different sets of co- 
ordinates, and that each energy must be necessarily positive. 
If this is granted, the necessity for equipartition of energy 
between the different waves follows, and in a state of maxi- 
mum entropy the total radiant energy must always be 
proportional to the temperature. 

This establishes the main proposition of the present paper. 
It may in addition be of interest to examine in detail ihe 
form assumed by the general argument when it is applied 
to the special problem under discussion. 


Special Investigation of Wave-motion. 


8. The system to which we shall now confine our attention 
will consist of a volume of ether in which a very small 
amount of matter is embedded, the function of the matter 
being solely to make possible the transfer cf energy in the 
eether between vibrations of different wave-lengths. Let 
there be supposed to be n vibrations in the ether, and let 
the co-ordinates of the sth vibration be Q,and R,. Let the 
number of additional degrees of freedom introduced by the 
presence of the matter be m, and let a typical one of these 
co-ordinates be §... 

We shall suppose that 2m, the number of co-ordinates 
associated with the matter, is infinitesimal in comparison 
with 2n, the number associated with the ether. It will also 
be assumed that m is so small that, for all configurations 
which are of importance, the energy residing in the matter 
is negligible in comparison with that residing in the ether. 

For this system equation (1) assumes the form 


n 0Q. OR, aa 2m OS. 


1 

In this equation the number of terms on the right is small 
compared with that on the left. If all matter were entirely 
absent the terms on the right would vanish altogether, and 
since the waves of different periods would then become 
independent dynamical systems, the terms on the left would 


1 


950 Prof. J. H. Jeans on Non-Newtonian Mechanical 


vanish separately. It follows that when the matter is suffi- 
ciently reduced in amount, the value of each term on the 
left is infinitesimal *. 

We accordingly suppose as an approximation that 


dQ, dK, 

sur he Lo 
for each value of s and proceed to examine the nature of the 
co-ordinates Q, R,. Equation (10) is the condition that 


Q.dR — RaQ shall be a perfect differential. Calling this 
dds, we have 


0¢, i 7 ee O¢, 
si? & +55.) | 


The rate of change of ¢, is given by 


ONG ee Oe Oe 
$, = 5Q, % +t Ok, R,=0 


by equations (11). Thus ¢, does not vary with the time. 
The energy E, of the sth vibration is some function, at 
present unknown, of Q, and R,. Its rate of change is 


_ OH, 04, _ ob, 09, 
~ 0Q, oR, AR, dQ; 
aud 0(k,, ¢,) : 
ci 0(Q,; R,) 
Since the energy mast remain unaltered with the time, 


the Jacobian must vanish, so that ¢, must be a function 


of E,. 


* An exception would occur if the matter were arranged so as to have 
free periods of its own, so introducing resonance effects ; then the right 


hand of (9) might be mainly balanced by a few terms only on the left — 


hand. But even if this is the case, there is no difficulty ; we can confine 
our attention to waye-lengths for which the resonance effects are 
negligible. 


Systems, and Planck’s Theory of Radiation. G55 
If 8 stands for 0¢,/OE;, a constant, the equations (11) 


become 


S OH, . OE; 
ee ap nok ty > 


in which the analogy with the Hamiltonian form is obvious. 

9. On giving ditterent values to @ and E as functions of 
Q and R we obtain different systems of equations of motion. 
Some of these may of course be incapable of representing 
wave-motion at all. 

The simplest form which can be given to ¢ is that of a 
linear function of Q and R, and this may without loss of 
generality be taken to be AR. The equations of motion 
become 


Q as aaa R = 0, 
of which the integrals are 
R=ceons.; Q = B—At. 


These will represent wave-motion if Q, R are taken to be 
phase and amplitude or phase and energy respectively, but 
will not satisty the condition of the co-ordinates being deter- 
mined uniquely from the state of the system, unless we 
suppose the space limited to a range 27 in the values of Q. 

10. The next simplest form which can be given to ¢ is 
that of a quadratic function of Q and R, and this may 
without loss of generality be taken to be £(CR?+ DQ’). The 
equations of motion become 


ORS RE Oy ay) wn santa) 


of which the integrals represent simple harmonic motion. 

11. Equations (11) show that the motion of the fluid in 
the generalized space is the steady motion of a homogeneous 
finid along the system of stream-lines @ = constant. For 
this to be capable of representing wave-motion the curves 
@ = constant must be a series of closed non-intersecting 
curves. The mass of fluid and system of co-ordinates may 
now be distorted so that these (or rather their projections on 
the plane Q,, R,) become a system of concentric circles, and 
this may be done in such a way that the fluid remains homo- 
geneous. On taking new axes the equations of motion 
become identical with (13), so that to represent wave-motion 
the co-ordinates must become identical with the Lagrangian 
co-ordinates and momenta. 

It follows that, however far removed the general equations 


952 Prof. J. H. Jeans on Non-Newtonian Mechanical 


of matter and ether may be from the standard form, yet 
when the matter is made to diminish indefinitely in amount, 
we may sup; ose, without any loss of generality, that the 
equations of wave-motion are of the standard form, and that 
the co-ordinates Q, R are the Lagrangian co-ordinates and 
momenta. Equipartition of energy follows as a direct 
consequence. 

12. In the general analysis of §7 it was assumed that the 
sesh of the generalized space was filled with fluid. The 
fluid must, however, be excluded from any parts which 
represent phy sically ‘impossible configurations, and if these 
parts are of sufficient extent, the exclusion of fluid may affect 
the final result. Let us examine whether anv arrangement 
of fluid can be found which shall so modify the result as to 
change the law of equipartition into the widely different law 
of Planck. 

Let us consider N vibrations having frequencies differing 
only infinitesimally from 27v. Their ‘total energy KH must, 
according to Planck’s law, be given by 


Nhy 


ae 


where / is Planck’s constant. Eliminating the temperature 


between this and equation (6), which is true no matter what 
parts of the generalized space are excluded, we obtain 


0, bos oie g(1+ 42”) 


J 


This gives on weird 


OR ta | 
8 =k ( N+ ,) ton (N+ 5, ) — jaloa;, f beens 


+ terms coe from the other vibrations. . (15) 


Let W be proportional to the volume of that region of the 
generalized space (less the excluded parts) in which B lies 
between E— ge and E+ Je. 

On comparison of equations (4) and (15) we have 


Siar cn! E E Bak 
log W = (N+ ile (N+ ww) jy 8 hy + cons. 


If we write P for E/hy, and use Stirling’s approximation, 


Att) 


Systems, and Planck's Theory of Radiation. Opa 


this becomes 
N+P ; 
W=C as CA Nt aie 
where C is a constant. This is of course Planck’s formula 
obtained by working backwards from Planck’s law. What 
is important is that (16) follows inevitably from (14); in 
other words, formula (14) can only be true in a generalized 
space in which the regions excluded are such that the 
remaining volume is given by (16). Furthermore, the 
necessity ‘for an indivisible unit of energy follows inevitably 
from (16), for Planck’s assumption of this indivisible unit is 
known to lead to formula (16), and there can be only one 
way of distributing the fluid in the generalized space so that 
W is a given function of E for all values of E. 

The analysis has, however, shown that the truth of P lanck’s 
law requires something more than appeared in Planck’s 
original papers. It is now apparent that it is not enough 
to postulate systems of vibrators capable only of holding 
definite multiples of a fixed unit of energy ; we see that the 
energy in the ether itself must also be atomic. Moreover, 
it is not sufficient that the energy should always in nature 
occur in complete atoms; what is required is that it should 
be physically impossible to divide these atoms. Jor instance, 
the requirements of this condition are not met by imagining 
a system of radiators which always give off energy in 
complete units; we must also have an ether structure such 
that no vibrations can possibly exist in it except in atomic 
amounts. If it is agreed that these conditions do not hold 
in nature, then we are driven to supposing that the state of 
the ether represented by Planck’s law is not a final steady 
state, or in other words that there is not thermodynamical 
equilibrium between the matter and the different vibrations. 

13. In conclusion, it may be worth noting an alternative 
method of arriving at Planck’s law. 

Other things being equal, if a vibration can have energies 
0, e, 2e,..., then the ratio of the probabilities of these events 
as in the usual gas-theory calculations, is 

D eee se cas 


where h=1/2RT, or, replacing h by its value, 
Ds aye Resa ace tyhad Oe aati coe oop 


If out of N vibrations under consideration, M have zero 


Phil. Mag. 8. 6. Vol. 20. No. 120. Dec. 1910. okt 


954 On Non-Newton'an Mechanical Systems, Se. 


energy, then the number which have energy e is Me-*/#?, 
the number having energy 2¢ is Me~?/8?, and so on. Thus 


N= M14 U8 eet |S Mi erate 
and if E is the total energy of the N vibrations, 
E = Me(e-*/2T + Qe-29/RT 4 |.) 
= Mee-@/RT/(1 —e-@/RT)2 


Ne : 
= ef/RT _ J? . . . ° . e . . ° (18) 


which gives Planck’s law on taking e=hv. 

It will, I think, be found that this caleulation of Planck’s 
formula is based on exactly the same physical ideas as those 
of the original theory of Planck. One essential and necessary 
feature of the theory is that it supposes the unit of energy 
for vibrations of moderate wave-length to be so great that 
the chance of a vibration having even one unit of energy is 
very slight. We notice that only a fraction (N—M)/N, or 
e-©®*, of the total number of vibrations possess any energy 
at all. At wave-length Amsx., ¢/RT=4:965, so that only one 
wave-length in 140 possesses any energy. At wave-length 
one-half of this the proportion is about one in 20,000. 

14. An interesting question is whether, if this theory is to 
be accepted at all, it ought not also to be expected to account 
for the failure of certain other degrees of freedom to receive 
the share of energy allotted to them by the theorem of equi- 
partition. Many types of motion, such as the internal 
vibratins of the atom, and the rotations of atoms or mole- 
cules, must have direct interchange of energy with the ether 
vibrations, so that if the latter are in temperature equilibrium, 
the former might be expected to be so. A rough estimate 
of the energy possessed by such degrees of freedom is 
furnished by the values of the specific heats. For a degree 
of freedom which has one thousandth of its equipartition 
energy, ¢/RT must be about 9:1, and only one degree of 
freedom in 9100 will have any energy at all. This result, 
when applied, for instance, to the rotation of the atoms of 
mercury vapour, is somewhat startling. 


Cambridge, Aug. 17. 


CVII. The Volatilization of Radium Emanation at Low 
Temperatures. By R. W. Bovis, PA.D., 1851 Exhibition 
Science Scholar, MeGull University *. 


Introduction. 


| aa the researches of Rutherford t and of Gray and 

Ramsay f we now have definite knowledge concerning 
the condensation and volatilization of radium emanation at 
temperatures higher than — 127° C. These researches, 
which have been earried out with quantities of emanation 
as large as are available, have given definite values of the 
vapour pressures corresponding to certain fixed temperatures. 

At the suggestion of Prof. Rutherford the writer has 
recently been investigating the volatilization of this emana- 
tion at temperatures from —1380° C. upwards. It has been 
necessary to adopt different experimental methods to suit the 
widely difterent quantities of emanation employed. 

When extremely small quantities of emanation, say of the 
order of the equilibrium amount from 0-001 mem. of radium, 
are coudensed upon a surface, one can no longer speak of the 
emanation being in a “liquid” or ina “solid” state. For, 
in these circumstances, the condensed “layer” must be of 
less than molecular thickness§$, and it would hardly be 
expected that volatilization would proceed in full accordance 
with the vapour-pressure laws as ordinarily understood. 

For experimenting under these conditions the method 
devised by Rutherford and Soddy || in 1903 is best ap- 
plicable. It will be recalled that the method was to condense 
from a current of gas, acting as carrier, a very small quantity 
of emanation upon the interior surface of a spiral tube, which 
was immersed in a bath at very low temperature. After- 
wards, while allowing the temperature of the bath to rise 
slowly, a very slow gas current was sent through the spiral. 
The emanation on volatilizing was swept out of the spiral 
and through an ionization vessel, where it marked its presence 
by causing an increase of ionization. 

The temperature of maximum volatilization was readily 


* Communicated by Prof. E. Rutherford, F.R.S8. 

+ Phil. Mag. (6] xvii. p. 723 (1909). 

{ Proc. Chem. Soc. xxvi. p. 82 (1909); and later, Journ. Chem. Soe. 
xev. p. 1073 (1909). 

§ Based on the determination by Rutherford, later substantiated by 
Gray and Ramsay and by Debierne, that the equilibrium amount of 
emanation from | gm. of radium has a volume of 0°6 cub. mm. at 
208 il a 

|| Phil. Mag. [6] v. p. 561 (1903). 


ov 4 


956 Dr. R. W. Boyle on the Volutilization of 


obtained by noting the temperature of the spiral when the 
ionization attained its maximum value. 

Using this method with condensing spirals of copper tube, 
Rutherford and Soddy* found that the volatilization of these 
small quantities of emanation was rather sharply defined, 
within a range of a few degrees about the temperature of 
—150° C. More recently. Mons. Laborde t applied the 

same method, using condensing spirals of different materials, 
and as a result claimed to have found marked differences in 
the temperatures at which the emanation will volatilize from 
the surfaces of different materials. Thus, from surfaces of 
iron, tin, silver (a silver tube), and of copper, the volati- 
vation 3 is claimed to take place at —155+2° C.; from the 
surface of a silver deposit on glass, —175+ 2° ©, ; and from 
the surface of glass itself, —177-4+2° C. It is remarkable 
that there should be a difference of 20°C. between the 
temperatures of maximum volatilization from the surface of 
a silver tube and from the surface of a thick silver deposit 
on glass; and it is noticeable that these temperature dif- 
ferences were only found where glass spirals were concerned 
in the experiments. 

These results had an important bearing upon the problem 
of the writer’s investigations, and in consequence some ex- 
periments were performed to examine the effect of glass and 
metal surfaces upon the temperatures in question. 


Method of Experiment. 


The arrangement of apparatus.is represented in fig.1. The 
condensing spiral DSD’ was of tubing 0°35 cm. diameter, 
and was immersed in sufficient pentane to cover the spiral in 
a test-tube of 3 cm. diameter. Usually there were ten or 
twelve turns in the spiral. 

The test-tube was immersed, almost the whole of its length, 
in a Dewar cylinder of 4 cm. diameter filled nearly to the 
top with liquid air. The spiral and the electroscope KK’ 
were “short-circuited” by the tubes Zand F respectively, 
so that a current of air could be used to sweep the conducting 
tubes free from uncondensed emanation without disturbing the 
condensed emanation in the spiral, and without unnecessarily 
contaminating the electroscope. 

The operation of an experiment was as follows :—Hmana- 
tion, which cd been mixed with air and stored in the 


TAP Es 
+ Le Radium, vi. p. 289 (1909). - 


Radium Emanation at Low Temperatures. 957 


receiver A, was forced slowly — usually about 0-4 c.c. 
per sec.—over the path ABCDEFG (see diagram). 

Carbon dioxide and water vapour were removed in the 
tubes B and C containing calcium chloride and soda-lime 
respectively. Condensation of the emanation from the 
purified current of air took place in the spiral S cooled 


Fig. 1. 


Thern{o>Gouple. 


Fvom Mavi olte 
Flask, 


AA | 
—> b.- 


So or 


HCO) oF 


i! 


Ht 
| 


| 


| 
! 


Il 
il 


Wi 
HANA ANIAN 


| 
| 
1 


! 


i 


nearly to liquid air temperature. The uncondensed emana- 
tion passed to the open air outside the laboratory building 
through the tubes F and G. After condensation of the 
emanation a current of air (7-5 c.c. per sec.) was blown for 
several minutes from a Mariotte flask, through the calcium- 
chloride tube X and the soda-lime tube Y, over the path 
XYZEFG. This air current carried outside the building 
the emanation remaining about the tubes leading to the 
electroscope. 

Next, a very slow and constant current of air from the 
Mariotte flask was sent through the condensing spiral over 
the path XYDEFG. This current was made very slow in 
order for it to attain quickly the temperature at any point of 
the:spirale - ~--- ya; iiehcrg Undiny 


958 Dr. R. W. Boyle on the Volutilization of 


Throughout the above operations the Dewar cylinder was 
kept nearly full of liquid air, but immediately they were 
finished the cylinder was raj idly drawn away from the 
pentane bath, quickly emptied of its liquid air contents, and 
then set back in place over the bath. ‘The stop-cocks of the 
electroscope were then opened. Under these conditions the 
temperature of the bath rose very slowly—about 0°4 C. 
per minute at the bottom. 

As soon as the electroscope stop-cocks were opened there 
was a slight rise in ionization. This was due to slight 
amounts of uncondensed emanation not completely expelled 
from the spiral and the conducting tubes. ‘This initial rise 
was in no way connected with the relatively large increase 
in ionization which marked the volatiiization at the tem- 
peratures presently shown. 

As already mentioned, pentane was used as a temperature 
bath. The pentane employed became very viscous at about 
—150° C., and consequently could not be stirred below this 
temperature. On this account the temperature of the bath 
was not uniform throughout but increased gradually from 
the bottom to the top. 

Under these conditions the temperature required must be 
that of the coldest part of the condensing spirals, namely, 
the bottom coil. This must certainly be “the case, for the 
writer has found from a few special experiments in which 
emanation was condensed in U-tubes, that condensation takes 
place in the limb of the tube through which the air current 

carrying the emanation enters. (The fact was ascertained 
by noting that the phosphorescence caused by the emanation 
and active deposit was confined to this limb of the U-tube.) 
Sinnlarly with spirals : condensation takes place in that half 
of the spiral through which the air current enters. If, when 
the process of warming has started, any emanation volatilizes 
in the upper and warmer coils of the spiral, the air current 
bears it to the lower and colder coils, where it immediately 
recondenses provided that the temperature rises slowly. It 
is therefore necessary that the air current should be very slow, 
and that the spiral should rise very slowly in temperature. 

For the above reason, the temperature of the bottom coil 
of the spirals was measured in all experiments ; but oppor- 
tunity was taken whenever possible to probe the other parts 
of the bath with a thermo-couple in order to acquire some 
information regarding the temperature distribution. 

The temperatures were measured by means of thermo- 
couples; the sensitiveness of the two element copper- 
constantan set, used in most of the experiments, was, on an 


Radium Emanation at Low Temperatures. 959 


average, 5 millimetre scale-divisions per degree centigrade. 
A calibration curve was constructed by means of the fixed 
temperatures of melting ice (0° C.), a mixture of solid COs, 
and ether (—79° C.), boiling ethylene (—103° C.), boiling 
methane (—164° C.), and liquid air of which the percentage 
of oxygen was determined by analysis (—186° C.). Some 
of the calibration curves were checked by further deter- 
minations from the melting-points of pure ether and of 
ethylene, but these points were difficult to fix with accuracy. 

At the beginning of the work an iron-nickel couple with 
a Gambrell moving-coil galvanometer was employed, but 
occasion arose to change these for couples of copper- 
constantan with an Ayrton and Mather moving-coil gal- 
vanometer. The wires of the thermo-couples were No. 30 
double-cotton covered, and the junctions were bare, without 
any protective covering, in order to avoid an appreciable tem- 
perature lag. Most of the present results were obtained with 
the warmer junction of the couple at the temperature of 
melting ice. A few others were taken with this junction at 
ordinary room temperatures, for which the necessary correc- 
tions were obtained. 

Instead of an electrometer as detector of ionization, an 
electroscope of low cubical capacity was employed. The 
degree of sharpness of the volatilization temperatures can 
be the better detected the smaller the ionization chamber. 
In consequence, the ionization chamber was merely a small 
air-tight brass tube, 11:0 cms. long, through which ran an 
insulated brass rod carrying at the end outside the ionization 
chamber a gold-leaf system. The volume of the free space 
in this chamber was about 6 c.c. 

In some of the experiments where metal spirals were used, 
the tube forming the spiral did not rise above the surface of 
the pentane bath. In these cases the conducting tubes D 
and D’ (see diagram, fig. 1) were made of glass and were 
joined to the tube of the spiral by rubber connexions, which 
experience proved to be quite satisfactory. This arrangement 
prevented the conduction of heat to the pentane bath along 
metal conductors, and consequently ensured a slower rate of 
rise of temperature. 


Discussion and Results. 

On a close examination of this flow method of experiment, 
it should be remembered that the temperature corresponding 
to maximum tontzation in the testing vessel is the temperature 
at which the rate of volatilization of emanation from the con- 
densing surface begins to decrease very rapidly to cero. 


ES ee TE 


960 Dr. R. W. Boyle on the Volatilization of 


It is known that when emanation is condensed on the 
inner wall of a vessel at any temperature, even down to the 
liquid air temperature, the emanation can be practically all 
removed by continuously pumping. The fact shows that 
there is an appreciable vapour phase of the emanation at 
all temperatures down to the temperature of liquid air. 
Consequently, in an experiment by the flow method, if the 
emanation be at first condensed, and the temperature be kept 
constant while the gas current is allowed to flow indefinitely, 
the molecules of emanation in the vapour phase will become 
entrained in the gas current and be removed. More mole- 
cules will then volatilize into the vapour phase from the 
condensed layer, and these in turn will also be carried away. 
This process will continue until all the emanation has been 
removed. It follows, therefore, that with a temperature not 
fixed but gradually rising, and with a constant gas flow, 
the temperature at which all the emanation has just been 
removed from the surface of the condenser will depend on 
(1) the quantity of emanation condensed, (2) the rate of rise 
of temperature. Experiments by the flow method to com- 
pare the influences of different surfaces on the process of 
volatilization should be carried out with the conditions 
regarding (1) and (2) at least approximately alike *. 

The quantities of emanation employed in the experiments 
varied between the equilibrium amounts from 2x 10-4 to 
2x 10-* mgm. of radium. Experience showed that the rate 
of volatilization at the final temperature is increasing so 
rapidly that variations in the quantity of emanation, much 
larger than the above, give very similar results. The range 
of accuracy in the experiments is considered to be +2° C. 

Curves I. and II. of fig. 2, which represent the connexion 
between ionization and temperature for spirals of lead and 
of glass, are typical of the behaviour for all the substances 
examined. The dotted curves represent the rise of tempe- 
rature at the bottom of the bath with time. 

The maximum ordinates GH and IK of the ionization 
curves give the temperatures at which the volatilization 
of emanation from the condensing surface very rapidly 
diminishes or ceases altogether. We may say that at these 
temperatures the emanation is all removed from the surface. 


But the ordinates, such as AB, CD, EF, and also MN, 


QR, show that at still lower temperatures there is appreciable _ 


volatilization. This could no doubt be noticed down to 
liquid air temperature if the electroscope could be kept from 
* His not meant here that the quantity of emanation cannot be varied 


from one amount over a range of ten or twenty times that quantity, but 
it should not be varied over thousands or tens of thousands of times. 


Radium Emanation at Low Temperatures. 961 


contamination by the first sweeping through of uncondensed 
emanation. With still smaller quantities of emanation this 
gradual volatilization as the temperature rises is more easily 


Fig. 2. 
[F400 Z 


ae. 
a 
1200 2 | 
F200 coun 


wo , 
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3 2 ‘ “Ry 
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r S : Ha 
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TEMPERATURE °C. 


noticed, for then there is not so much uncondensed emanation 
to contaminate the electroscope. With larger quantities of 
emanation the rise of ionization to a maximum appears all 
the more sudden, because the amounts of emanation vola- 
tilizing at the lower temperatures are relatively much smaller 
than the amounts at the temperatures corresponding to 
maximum ionization. 

Experiments by another method will be described in a later 
paper showing that a gradual and continuous volatilization 
can be detected from —180° C. upwards. The fact that the 
maximum ordinates of curves I. and II. do not fall at the 
same temperature is explained later. 

The results of a number of experiments, using spirals of 
the materials named and of diameter of tube 0°35 cm., are 
given in the following tables. In these experiments, attempts 


r 


zation o 


l 


Dr. R. W. Boyle on the Volatil 


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Radium Emanation at Low Temperatures. 963 


were always made to keep the conditions regarding (1) the 
rate of flow of air-current, (2) the rate of rise of temperature, 
and (3) the quantity of emanation condensed the same for all 
materials. Experience showed that there could be consider- 
able latitude in all three factors without obtaining final 
temperature results outside the error of experiment. This 
is especially so with regard to the quantity of emanation. 

The numbers tabulated under ‘ Maximum Ionization,” 
representing divisions per minute in the electroscope, give a 
rough approximation of the relative amounts of emanation 
used in the different experiments; those under ‘“ Rate of 
Rise of Temperature” refer to the bottom temperatures of 
the bath at the time of maximum ionization. The “ Rates of 
Air-Flow ” refer to a temperature of —160° C. 

The temperature correction for the time taken by the 
volatilized emanation to pass from the condensing spiral to 
the electroscope was negligible in comparison with the error 
of experiment. 

It will be noticed from the diagram and table (Table I.) that 
in the case of glass the temperature was consistently a few 
degrees lower than in the case of metals. The difference in 
the case of silvered glass was not so large, and in consequence, 
the smaller heat conductivity in glass, compared with metals, 
was not quite satisfactory as a reason for the differences 
shown. ‘To obtain more information on the point further 
experiments were performed. In these, condensation of the 
emanation took place in small glass and lead spirals of only 
two or three coils, the planes of the coils when placed in the 
bath being vertical instead of horizontal, and the spirals being 
just covered with pentane. Under these circumstances there 
could only be very little temperature variation over the spirals. 
The thermo-couple (a single copper-constantan element) 
was threaded through the spiral tube so that the temperature 
on the inside of the tube was the one determined. The wires 
of the thermo-couple were brought to the open, outside the 
bath, through the walls of the glass conducting tubes, and 
the holes through which the wires emerged were closed by 
sealing-wax. Taking these precautions, the final temperatures 
with both lead and glass spirals come, within the experi- 
mental error, to the same value, as can be seen from the table 
w hich follows. 

This result, taken in conjunction with what has gone before, 
has one of two explanations. Hither the condensed emana- 
tion volatilizes from the surfaces of glass, thermo-couple wires, 
and of lead at the same temperature ; or, some emanation 
remains condensed upon the surface of the thermo-couple 


of 


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Dr. R. W. Boyle on the Volat 


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Radium Emanation at Low Temperatures. 965 


wires at a temperature a few degrees higher than that at 
which the emanation volatilizes from glass. 

In any case it can be said that the temperature differences 
in the pr resent experiments were small. The emanation could 
not have volatilized from the glass surfaces at temperatures 
as low as —177°C., for then the volatilized emanation would 
have shown its presence in: the ionization chamber imme- 
diately after opening the stopcocks, and without any appre- 
ciable wait for the temperature to rise. Moreover, in some 
experiments condensation only took place at — 177° o 
— 178°C. 

Mention has been made before of attempts to get an idea 
of the temperature distribution in the bath by means of a 
second thermocouple. It was found that no matter how the 
temperatures throughout the rest of the bath varied, the 
bottom temperature at maximum volatilization of the emana- 
tion was about the same in all cases. The temperatures at 
the top and middle parts of the bath would of course depend 
on the depth of pentane in the bath, an average value of the 
temperature gradient being about 2° C. per centimetre depth. 

It was not considered necessary to experiment with iron, 
tin, and other metals, since M. Laborde obtained the same 
result with these as with copper and silver. 

No doubt the above volatilization temperatures could have 
been fixed with greater precision if a bath which remained 
liquid over the required range of temperature could have 
been used. But it was not feasible to procure the large 
quantities of a liquefied gas (such as ethylene) and of liquid 
air, which would be necessary for such a large number of 
experiments. The experiments of M. Laborde must have 
suffered from this same disadvantage, for it is not likely that 
a metallic bath of granulated copper would be of uniform 
temperature throughout. Some trials by the writer, dupli- 
cating as nearly as possible the conditions, have shown that 
after the process of warming from the liquid-air temperature 
sets in, there is, as one would expect, a continuous and large 
temperature gradient from the bottom of the bath to the top. 
A metallic bath also is of a discontinuous structure, and 
allows the penetration of air down to the bottom as soon as 
the liquid air has evaporated. For these reasons the use of 
metallic copper as a temperature bath was abandoned in 
favour of pentane. 


Conclusion. 


In the series of experiments just described only small, and 
almost inappreciable. differences (4° C.) can be found in the 


966 Mr. T. H. Blakesley on the Diameter of 


temperatures at which small quantities af radium emanation, 
when once condensed, will volatilize from the surfaees of 
glass and of metals. With this flow method of experiment, 
using quantities of emanation varying between the equili- 
brium amounts from 2x 10-4 and 2x 10-8 mem. of radium, 
and with a rate of rise of temperature of about 0&4 C. per 
minute, it was found that there is a slight and gradually 
increasing volatilization until the temperature approaches 
—160° GC. Approaching this temperature, the volatilization 
becomes relatively very rapid, and above it volatilization 
practically ceases. 

The temperature, —160° C., mentioned above, is merely 
the final temperature of separation of the emanation from the 
condensing surfaces in the experiments ; its measurement 
serves as a method of comparing the effect of the various 
surfaces mentioned on the volatilization. This temperature 
is some degrees lower than the majority of results in the 
original experiments of Rutherford and Soddy, where the 
temperatures measured were the average temperatures over 
the whole of the condensing spiral, and where there was a 
higher rate of rise of temperature. 

It is not to be understood that it is impossible for emana- 
tion to remain condensed on a metallic or glass surface at 
temperatures higher than —160° C.; for further experi- 
ments by another method have shown that the temperature 
of final separation from the surface depends, as would be 
expected, on the quantity of emanation condensed. These 
experiments will be described in a later paper. 

The writer is glad to acknowledge great kindness and 
assistance on the part of Prof. Rutherford. 


University of Manchester, 


August 1910.  f am 


R: ’ 
CVII. A Means of Measuring the Apparent Diameter of the 
Pupil of the Eye, in very feeble Light. By T. H. 
BLAKESLEY *. 


FP\HE advantage obtained by the magnifying power of a 

telescope or a microscope may be very seriously 
diminished if the orifice of the eyepiece through which the 
light issues to the eye is so small as not to fill the pupil with 
licht. The magnification is inversely proportional to the 
focal length of the eyepiece, and therefore to the orifice 
with which it is furnished, which therefore has a smaller 


* Communicated by the Author. 


the Pupil of the Kye in the Dark. 97 


diameter the greater the magnifying power, as depending 
upon the eyepiece. 

If the object in view is a very faint star or nebula in the 
case of a telescope, or an ultra-microscopic subject in the 
case of a microscope, the field is necessarily very dark. All 
side light must be carefully avoided, and the pupil of the 
eye is, without doubt, in a highly expanded condition, so 
that the eyepiece employed may have an orifice smaller than 
the pupil. The object of this communication is to show how 
the diameter of the pupil may, with fair accuracy, be measured 
in such all but complete obscurity. 

The accompanying sketch will illustrate the means employed 
to effect this. 


The apparatus consists of what is virtually one tube of a 
common opera-glass, with the eyepiece removed. Into the 
sliding tube is inserted another tube B of suitable length, 
and this is pushed home until it is stopped by the offset at 
the far end of the sliding tube. With the apparatus at its 
longest the distance from the open end to the near side of 
the object-glass may be measured, and also the exposed 
portion of the sliding tube. In any other position the defect 
in the latter is equal to thatin the former, the value of which 
is thus at any time readily determined. 

In the cell of the object-glass and in contact with it on its 
near side, is inserted a screen made of the black opaque 
paper used by photographers. Two small pinholes PP’ are 
made upon one diameter, at equal distances from the axis 
AF of the lens. IF is the principal focus of the lens. The 
focal distance AF requires to be known, and it is a quantity 
which can be determined with considerable accuracy. 

The instrument is directed upon a distant and solitary 
light when the night is dark. A pair of fine beams will 
then pass through the pinholes PP’, and converge to the 
focus F. 

These will both be visible to an eye placed coaxially with 
the lens, if the apparent diameter of the pupil of the eye is 
not less than the distance apart of the beams in the position 
along the axis which the apparent pupil happens to occupy. 


968 The Diameter of the Pupil of the ye in the Dark. 


Let E be a position in which this is just possible, E being 
nearer to the object-glass than is F the principal focus. 
Then if D denotes the distance between the pinholes, d the 
diameter of the apparent pupil, and v the focal distance AF, 
the following equation is true, viz. : 


a=~. 9D, 


In making this observation the eyelids must be held 
against the open end of the tube B, not merely with a view 
to exclude any external liyht but also to ensure a fixed 
relative position of the eye and the tube. 

The tube B is now withdrawn, and a second tube of the 
same calibre takes its place, of a length suitable for dealing 
with the second position E’ where the pupil will have a 
diameter equal to the distance between the beams of livht. 
A second adjustment is made resulting in the equation 


D 
d= —. FE! 
‘4 
Adding the two equations, 
D EE’ 
a ee 
pene: 


Now, as the two tubes employed have the same diameter, 
the position of the eye relatively to the ends of the tubes 
will be the same, or EE’ will be equal to the difference 
between the distances from the lens of the open ends of the 
two tubes when in adjustment. 

Denoting therefore these distances by lL and / respectively, 
the diameter of the apparent pupil is given by the equation 


The following observations muy be made. It is necessary 
on approaching adjustment that the eye should be fixed 
upon A, half way between P and P’. 

Let the diameter PP’ be supposed horizontal, and suppose 
the eye to be slightly nearer the object-glass than the 
position E. Neither beam is seen if the eye is directed to 
A, but if the eyeball is turned to the right, the pupil is 
shifted in the same direction, and P at once is seen. 

If the eye is just on the side of EK remote from the object- 
glass, both beams are seen when the eye is directed to A, 
but a turn of the eyeball to the right puts P! out of view. 


Study of Variable Currents by “Phaseograph.” 969 


If the eye is slightly nearer the object-glass than the 
position EK’, and directed to A, both beams are seen, but if 
the eyeball is turned to the right, instead of P (which appears 
to the right always) becoming more plainly seen, it 
disappears. 

And finally, if the eye is just on the side of EH’ remote 
from the object-glass, and directed to A, nothing is seen. 
But if the eye is turned to the right, P’, towards the left, 
becomes visible. 

These facts, obvious enough, are pointed out as assisting 
very considerably the process of adjustment. If difficulty is 
experienced in keeping the eye fixed upon A, a small pinhole 
may be made in the screen at that point. The beam of 
light through it will, during the processes described, be 
always visible. 

The values of the diameter of the apparent pupil, under 
the circumstances indicated, in the writer’s own case, have 
been found to vary from 6°74 to 7:20 millimetres, the higher 
values being less frequent. He cannot but look upon the 
value 4 of an inch (8°47 millimetres), quoted occasionally in 
text-books, as extremely abnormal. 


CIX. The Study of Variable Currents by means of the 
“ Phaseograph.” By MANNE SiEGBAHN*, 


[Plate XIX.] 


Introduction. 


N order to characterize the electric state of a conductor 
through which an electric current passes, the two 
variables, strength of current (2) and voltage (e), may con- 
veniently be made use of. As long as we have to deal with 
metallic conductors and constant currents, there is between 
these two variables the simple relation : 


~ = const. = m, aoa Cay Bake ee (1) 


where the constant m signifies the ohmic resistance in the 
conductor concerned. 

In the general case, variable currents and conductors of 
different kinds, the connexion between the variables is more 
complicated. In order to obtain a survey of matters in these 


* Communicated by the Author. 
Phil. Mag. 8. 6. Vol. 20. No. 120. Dec. 1910. 38 


970 Mr. M. Siegbahn on the Study of Variable 
cases one variable may be expressed as the function of the 
other, 

e= f(t)... + = 2. a 


For a long time we have used the graphic representation 
for the magnetic magnitudes, and thus learnt to master them. 
Also in a closely connected branch of electricity, i.e. the 
electric machines, the graphic method has been used to 
advantage. This has particularly been the case, since 
Hopkinscn’s epoch-making works made it possible for us to 
calculate “magnetic loops.” For alternate currents this 
excellent method has only recently begun to be made use of. 
It was not until Kauffmann * had proved its applicability to 
electricity passing through gases, and developed the conditions 
of stability, that it could come into proper use where alternate 
currents are concerned. 


l. Characteristics. 


If the function (2) is graphically represented, the so-called 
characteristic curve is obtained. We will now consider the 
most important qualities of these curves. 

Fig. 1 reproduces a characteristic curve or, more shortly, 


Fig. 1. 


0 


a characteristic. It gives for any i-value the corresponding 
value of e, supposing 2.¢. one has in every point to deal with 
a stateonary state. To exemplify our case we may take 


* W. Kauffmann, Ann. d. Physik (4) ii. p. 158 (1900). 


Currents by means of the “Phaseograph.” i 


the given curve to represent the characteristic of a carbon 
filament lamp. How is the resistance now obtained for a 
certain strength of current (i,)? The resistance is defined 
by the equation (1). Consequently the resistance graphically 
expressed is the tangent of the angle ({OA). If on the other 
hand the resistance is defined as 


de 
di’ 


another value would be obtained. Hence it follows that 
the function (1) used for stationary currents in metallic’ 
conductors, is of little value in more complicated cases. Its 
place is taken by the characteristic, where variable currents 
or inhomogeneous conductors are concerned. If the charac- 
teristics of two conductors.are known, the characteristic 
of a conductor composed of the two can be constructed from 
them. Here the following laws for coupling in series and 
parallel coupling of the two characteristics hold good. 


A. Coupling in Series.—For a certain strength of current (7) 
the two e-values are added. 

B. Parallel Coupling—For a certain voltage (e) the two 
2-values are added. 


Fig. 2 shows the graphical construction of the charac- 
teristics in question. 

If the voltage (E) and the -. characteristics are given, 

38 2 | 


972 Mr. M. Siegbaln on the Study of Variable 


we are in possession of the means necessary for the deter- 
mination of the e- and i-values. According to the additional 
theorem 


EB = f,(i)+f@yo.) «(1) eae 
R-AO) 4 h@e ie 


This alteration of our formula gives us a simple method of 
graphically finding the e- and 7-value. 


or 


Fig. 3. 


From the line e=H, /\(2) is set off: its intersection with 
f,(i) is the required point of equilibrium A. 

From this Kauffmann’s conditions of stability are easily 
found. From (3) is obtained by differentiation with respect 
to the strength of current (7), 


Of , Ofe. 5 
Jitfit aint ae 3 . + jn nn 


for a stable equilibrium, it is required that this expression 
shall be greater than the given voltage FE; or 


stable : af + vs >O0 «.\. .) 
indifferent : oh af Ole =0>~ > oy |. 
unstable : on + ws <Q... 4 
Graphically a signifies the angular coefficient of the 


tangent at the point in question. 


Currents by means of the “Phaseograph.” 973 


Il. Alethods of Registering Characteristics. 


We can distinguish between two different kinds of charac- 
teristics. 


A. Statical characteristics. 
B. Dynamic characteristics. 


A refers to stationary or quasi-stationary currents; these 
can be registered by ordinary voltmeters or ammeters. B, the 
dynamic characteristics, occur with variable especially alter- 
nate currents; these must be registered by instruments which 
at every moment: indicate strength of current and voltage. 
For this purpose various oscillographs with two movable 
systems are used: one registering strength of current, the 
other voltage. By the construction of the connected values 
in a co-ordinate system the required characteristic is obtained. 
Consequently, this is practically an indirect method, which 
besides implies much waste of time. Another ditficulty also 
arises which makes the results uncertain, 7.e. the defective 
adjustment. The adjustment is effected by directing the 
reflexions from the mirrors of the two movable systems 
towards the same point. If these reflexions were, let us say, 
geometrical circles with the same intensity of light, the 
adjustment would, no doubt, be possible, but as they present 
irregular figures, it can be done only with a limited degree 
of accuracy. ‘To this may be added also the cir cumstance 
that the adjustment sometimes is altered while the instrument 
is registering. On another occasion I will deal more closely 
with this question. ‘To avoid the difficulties just mentioned, 
particularly the tedious redrawing of the curves, the two 
movable systems must in some way be made to co-operate. 
The apparatus I am going to describe is based on this idea. 


II. Lhe Phaseograph. 


In a communication to the Physikalische Zeitschrift * I have 
demonstrated the chief features of the construction. As 
shown by fig. 5 (Pl. XIX.), the apparatus consists of two 
parts: the electromagnet and a box of brass with an ebonite 
or slate lid. Im the first construction the electromagnet was 
mace of two soft iron bows, each bar being provided with a 
solenoid. This arrangement, however, proved useless. Of 
the tour pears fields required, only two were sufficiently 
strong, i.e. those formed between the two bars of the same 


>? 
bow. Ly the final construction the electromagnet - was: made 


* M, Siegbahn, Phys. Zeits. x. 1909, p. 1017. 


974 Mr. M. Siegbahn on the Study of Variable 


of a soft iron ring with four upright iron rods. Hach of 
these is covered with a solenoid of copper wire. The solenoids 
have been arranged so as to close to 80 volts if coupled in 
series. In the brass box are fixed four pole-pieces of the 
shape indicated by fig. 4. Each of these is provided with a 


Fig. 4. 


cylindric plate which fits one of the four electromagnets. In 
the magnetic fields NS, N’S' are extended two thin silver 
wires (diameter 0°02 mm.), AB and CD. The tension in 
these wires can be regulated by means of special screws. The 
larger screws will be used for turning on the electric current. 
At the crossing-point the two wires are insulated with a plate 
of mica, and also provided with a small mirror. 

If a constant current is made to pass through one wire, 


f.i. AB, one-half will be lifted, owing to the magnetic field, 


the other will be pressed down; hence the mirror will be 
turned at a certain small angle. With sufficiently small. 
angles the turning is proportional to the strength of current. 
Owing to the great tension in the wire, the new equilibrium 
is not restored until after a considerable time of oscillation. 
If the box is filled with a suitable damping fluid, this move- 
ment can be made aperiodical. With the ordinary two- 
stringed oscillographs, vaseline-oil is used for this purpose. 
With this apparatus the vaseline-oil did not act asa sufficiently 
strong damper. Caster-oii, however, proved effective. 


Currenls by means of the ‘“Phaseograph.” 975 


There remains to be described the mounting and the special 
arrangements, as shown by fig. 6. 

The Nernst-stvle A throws a sharp 
light on a small diaphragm, which at 
exposure can beshut withashutter. By 
the aid of the two lenses, L; and I.., an 
image is formed on the ground glass- 
plate EF. For photographic purposes 
the latter can be replaced by a chassis 
with plate. The lens and the diaphragm 
are sheltered from outside light by a 
cardboard tube. 

If we want to study a current curve, 
i.e. use the phaseograph as an ordinary 
single oscillograph a revolving mirror 
with driver is put on, and the ground 
glass-plate is removed. The image then 
appears on the screen BC; the desired 
current curve is obtained by putting 
the mirror in rotation. 

By means of binding-screws and 
interrupters (8,23) on the outside of the 
box the electric current is conducted to 
the electromagnet (S,) and the two measuring-wires (8,83) 


IV. Other double Oscillographs. 


In this place it may be convenient to mention some other 
oscillographs with two co-operating systems, which were 
constructed about the same time as the one I have just 
described. 

According to a communication in Verhandl. der deutsch. 
Phys. Gesellsch. xi. (1909), A. Wehnelt has made use of 
two oscillograph-slides, one of which was introduced into a 
magnetic field, which was to be examined for the demon- 
stration of magnetization curves. At the request of Siemens 
and Halske, Berlin, Hansrath tried to use this principle 
for registering hysteresis curves. The method in this form, 
however, proved useless, and the instrument brought into the 
market was composed of an “ abstimmbar ” vibration galva- 
nometer on an ordinary oscillograph-slide. 

The * Hysteresigraph” constructed by H. Abraham served 
the same purpose. 

F’, Piola’s double oscillograph may also be mentioned. It 
consists of two turning-coils at right angles. Its use is con- 
fined to slowly variable fields. 


976 Mr. M. Siegbahn on the Study of Variable 


V. Theory of the Phaseograph.: 


In theory the phaseograph can be treated in the same way 
as the oscillograph. For the movable system the following 
equation is obtained :— 

KK, tA= +OS=F. « . ee 
dt? dt ( ) 
The letters signify : 

KX, moment of inertia ; 

A, damping factor ; 

(‘, directive force ; 

F, applied force. 

If a Fourier’s series is put for F, 


F= 3%, sin (kot—d,), . . .) 
| 
the integral to (9) will be 


1 Si | 
onan 1 a a RE a af 5s k t{— owe: . 3 1 
C2 7G — Rat)? date in hot — b2— Yas C1) 


A 2or Kk : 
Qakr 
tan Y= Jona: + it) 


If the mirror-oscillations are to represent the proposed 
Fourier’s series I’, the integral 3 for a=1: 
1 eo, bs 
must give the same curve as F. Any experimental test of 
this formula is almost out of the question. 


VI. Working of the Apparatus. 


When using the apparatus it is necessary to ascertain if 
the deviation is proportional to the strength of current. For 
this purpose the ground-glass plate was replaced by a glass 
scale, and simultaneous readings were made on this scale 
(from any 0-point) and an ammeter. The following values 
were obtained :— 


Currents by means of the “Phaseograph.” OF 


Streneth of current. Deviation, 
he 2. 
10 21077 ang. iaF 

90 | ok 2°49 
80 1°45 ade 
70 i298 PS) 
60 LiL0 aio 
O90. 0-94 areas 
AQ) 0-76 3°60 
30 0°58 ee 

0 0-00 4°48 


7 is noted when one measuring-wire only was used, 2 when 
the same current passed through both the wires. In the 
subjoined graphical diagram the direction of 2 is reversed. 


Vic. 


“I 


Mev. 


The diagram shows that there is proportionality between 
deviation and strength of current. 

If the instrument is used for registering characteristics, the 
method is obvious: we have merely to shunt part of the 
current off to one measuring wire, while through the other 
a current is passed that is proportional to the examined voltage 
Later on I will give complete coupling-diagrams for this use 
of the apparatus. 


Another way is to compare two alternate currents with 
each other, f. 7. strength of current and voltage in transformers, 
phase-differences between current and tension by self-induction 


978 Mr. M. Siegbahn on the Study of Variable 
and capacity, Through one wire passes then a current 


t=Hsin@t, 2 sin. |, 
and through the other 


e==e,sin(mi+¢d). » . < oo ee 


By the combined action of these oscillations the luminous 
point describes a figure the equation of which is obtained 
from (16) and (17) by the elimination of ¢: 

92 2 9p) 

i e 2ei ae 
= + —— cosh=sin' d. . in 
a 2 2 
by & Cyto 

In the general case the luminous point, consequently, 
describes an ellipse. What is of interest in this case is the 
phase-difference between the two currents. It is especially 
interesting as by the phase difference can be calculated 
e.g. the self-induction of a bobbin. For the phase-difference 
between current and voltage we have 
2a L 
fang == ——— 


. . . ’ . . 19 
wr. * (19) 
T, period ; 

L, self-induction coefficient ; 


w, ohinic resistance. 


phase-difference from the registered ellipse. 


Fig. 8. 


By letting the two currents (e and 2) register one at a 
time the resistance is altered, till the same amplitudes are 
obtained, 


gee 


Currents hy means of the “Phaseograph.” 979 
The equation (18) is then simplified to 
?P+e—Yeicosd=i, sin?d.. . . . (21) 


This equation represents ellipses inscribed in a quadrangle. 
Its axes are consequently the lines 


Bea eig Md hah Slee Ya) ee 


If e¢ is eliminated between the equations (21) and (22) we 
obtain 
ye eos =ty sie a ws) ey 


The two 7-values obtained from this formula are the coordi- 
nates of A and B. They can be exchanged for the semiaxes 


a. 0; ‘8 1 
Qty / 25 - C= tf 2 3 
Go 2h b= 2s 


a?(1—cos })=72,” sin? 
b?(1+cos 6 =7,? sin* sh 
By division is obtained 
a’—a cosd=b? +b’ cos ¢; 


a? —b? 


COs o= QBs : 


« C24) 

We have consequently only to measure the axes of the 
ellipse to find phase-difference and from this self-induction 
and capacity. 

There remains to be mentioned a third way in which the 
apparatus can be worked, 2. e. by the use of an auxiliary 
current in one measuring wire. In the first place we can 
then think of the use of a constantly decreasing or increasing 
current which gives to the luminous point a lateral deviation 
with constant rapidity, at the same time as the other iea- 
suring-wire registers the desired current-curve. Here belong 
also the methods of registering the characteristics of electric 
machines. In another place we will deal more fully with 
this subject. There remains to be considered some appli- 
cations of the above-mentioned methods. 


VII. Alternate Currents. 
Under this heading I will bring together some registerings 
of the relations between current and voltage with self-induc- 
tions, capacities, transformers, Xe. 


Figs. 9 and 10 (Pl. XIX.) show some current-voltage 


980 Study of Variable Currents by “Phaseograph.” 


diagrams with a self-induction bobbin ; in the latter case a 
soft-iron cylinder has been introduced into the bobbin. 

Figs. 11 and 12 (Pl. XIX.) show the current-diagram of 
the primary and secondary coils of a transformer according 
to the following coupling-diagram (fig. 13). 


Fig. 11 without iron core, fig. 12 with. In the phaseo- 
grams the higher oscillations of the alternate-current muchine - 
are clearly visible. 

We further subjoin, with no other comment, a phaseogram 
registered with a Wehnelt interrupter (fig. 14, Pl. XIX.), 
It shows that the apparatus can be used also for these rapid 
oscillations. 

To instance the method of calculating the self-induction 
from the phaseogram, we subjoin the following diagram 
(fig. 15, Pl. XTX.). for phase-difference we obtained above 
the following formula :— 


cosd= ——;—,, . ., i 


a . . 
where ;, expresses the relation between tle chief axes. In 
2 


the diagram 


from which is obtained 
h=37° dd’, 
The self-induction is L. 
@ 


L= 5x tan ¢. a (267 


w=ohmiec resistance=1 ohm ; 


— 9 
N=the frequency eS 


L=0-0043 henry. 


Positive Thermions emitted by Alkali Sulphates. 981 
VIL. Aluminium Cell. 


To show phaseograms by more irregular currents, I have 
made some experiments with an aluminium cell consisting 
of an aluminium wire against sheets of lead in dilute sulphuric 
acid (accumulator acid). 

The three phaseograms show the cell in various stages : 
fig. 16 soon after the closing of the current; fig. 17 when 
the current has passed for a while; lastly fig. 18 shows an 


inactive cell (PI. XIX.). \ =e 


1X. The Positive Thermions emitted by the Alkali Sulphates. 
By O. W. Ricnarpson, ILA., D.Se., Professor of Physies, 
Princeton Oniversity™. 
[Plate XX. | 
co by the author + and Mr. Hulbirt f have 

4 shown that the bulk of the positive thermions emitted by 
the commoner conductors, which can be heated to a sufficiently 
high temperature, possess very nearly the same value of e/m : 
a value which corresponds to a molecular weight m for the 
ions of about 25, on the assumption that they carry the same 
charge as the atom of hydrogen in electrolysis. There are a 
number of different ways in which it might happen that the 
various substances investigated might give rise to ions having 
the same specific charge, and these are discussed at length in 
the papers referred to. One of these views, and the one 
which seemed to have the most evidence in its favour, was 
that the ions consisted of atoms of sodium which were present 
as a common impurity in all the substances investigated. 
This view is supported among other things by the well-known 
fact that sodium compounds are very widely distributed, and 
by the fact that the value of m for the ions is very close to 
the atomic weight of sodium (23°05). 

It was this view that suggested the. present investigation. 
For it was felt, since the alkali metals and their compounds 
are so similar to one another in both chemical and physical 
properties, that if sodium compounds gave rise to positive ions 
having a certain specific charge, it ought to be possible to 
obtain ions having other values of e/m by substituting salts 
of the other alkali metals for those of sodium. The author 
therefore decided to measure the value of e/m for the positive 
ions emitted by the sulphates of the different alkali metals 


when heated. 
* Communicated by the Author. 
of Phil, Mag: ta vol xvi. p. 740 (1908). 
{ Phil. Mag. [6] vol. xx. p. 545 (1910). 


ee os 


G82 Prof. O. W. Richardson on the Positive 


The method adopted consists in measuring the deflexion 
of the path of the ions, between two parallel plates at 
different potentials, produced by a known transverse magnetic 
field. It was, in fact, the same as that used in the investi- 
gation of the specific charge of the ions emitted by hot bodies. 
So that, for the description of the apparatus and the method 
of using it, it will be sufficient to refer to the first of the two 
papers mentioned, which contains a very full account of the 
matter. 

The experiments were carried out in the following manner: - 
a strip of platinum 0°5 cm. long, ‘002 em. thick, and about 
‘05 cm. wide, was mounted in the apparatus. The distri- 
bution of the ionization over a parallel plate about *5 em. 
away was then determined in the usual manner, with 
oppositely directed magnetic fields of about 4700 lines 
perem. From the displacement of the maximum the value 
of e/m for the positive ions from platinum could be calculated. 

After the measurement had been made, the platinum was 
strongly heated for several hours until it gave only a 
negligible leak at the temperature at which the test was 
made. The apparatus was then taken down, and a layer of 
the pure alkali salt to be tested was melted over the front 
surface of the strip. The apparatus was again set up and 
exhausted. It was now found that a large positive ionization 
could be obtained at a temperature below that at which the 
test on the platinum had been made, showing that the 
ionization was due to the alkali salt and not to the platinum. 
The distribution of the ionization on the opposite plate was 
now determined with the magnetic field in opposite directions, 
and the displacement of the maximum measured. 

From this displacement together with the distance and 
difference of potential between the plates and the strength of 
the field the values of e/m and of m could be calculated from 


the formula : e/m = e _ In calculating the value of m 


the ions were assumed to carry the same charge as an 
atom of hydrogen in electrolysis, and the value of e/m 
for the hydrogen ions in electrolysis was taken to be 
9°66x 10? E.M. units. In the present case we have a 
valuable check on the results thus obtained, on account 
of the simultaneous measurements which were made of the 
disp!acement in the case of the ions from platinum. Previous 
work has shown that the value of e/m for the positive ions 
emitted by this metal* does not vary much for different 


* Richardson & Hulbirt, loc. cit. 


Thermions emitted by the Alkali Sulphates. 983 


specimens of the metal. By assuming the ions from plati- 
num to have the average value (25:5) previously found, it 
was possible to calculate the values of e/m and m for the salts 
investigated without making use of the linear dimensions of 
the apparatus. If we could rely on the constancy of e/m for 
the positive ions from platinum, we could in this way get rid 
of the uncertainty arising from the difficulty of knowing the 
distance between the hot strip and the plates accurately. It 
is probable, however, that the value of e/m for the positive 
ions from platinum is only approximately constant, so that 
this method can only be regarded as a check on the other. 
The disagreement between the two methods was not much 
greater than the errors of experiment. 

It has already been pointed out that the method is liable to 
an error on account of the uncertainty as to the position of 
the strip when heated. On account of the expansion and 
consequent curvature of the strip, this is not the same as 
when it is cold. This uncertainty was avoided in the former 
experiments by using the apparatus to measure the known 
value of e/m for the negative electrons. Assuming that the 
measurements would be liable to the same error whatever the 
sign or mass of ions, this enables us to determine a correction 
to be applied to the positive ions, 

In the present investigation this procedure has not been 
followed. The distances z which have been measured are 
those between the back of the strip and the front of the plates 
in the neighbourhood of the strip. The distances between 
the plane from which the ions are emitted and the slit-planes 
will, therefore, be less than this by the thickness of the 
platinum plus the thickness of the layer of salt. The former 
was ‘002 cm. and the latter about as much. The platinum 
strip was arranged so that it curved away from the plates 
when heated. It is believed that the increase of the distance 
due to the curvature more than counterbalances the effect of 
the neglected thickness. The effect of these errors in the 
distance is probably to make the values of m deduced from 
the observations too low. On the other hand, there is an 
intrinsic error, due to the apparatus not satisfying the theoreti- 
eal conditions, which tends to make m too high. It is believed 
that these effects just about balance one another when the 
value of z as determined above is substituted in the formula. 
At any rate, it is clear that however considerations of this 
kind may affect the absolute values of e/m and m, they can 
have very little influence on the relative values of these 
quantities. 


984 Prof. O. W. Richardson on the Positive 


Preliminary Experiments. 


On account of the very close similarity in the properties 
of the alkali metals and their compounds, we should expect 
that the sulphates of the different alkali metals would give 
rise to positive ions whose values of e/m and m would be 
determined by the atomic weight of the metal under 
investigation. This conclusion is to be expected whether 
the ions contain more than one chemical atom or not ; for 
it is to be expected, from the similarity of the properties 
of the alkali metals already alluded to, that the ions 
from the different metals will be similarly constituted what- 
ever their constitution may be. For example, if the ions 
from lithium sulphate were molecules of Li,SO, which had 
lost a negative electron, we should expect the ions from 
potassium sulphate to be molecules of K,SO, less a negative 
electron : if the ions from lithium sulphate were atoms of 
lithium less a negative electron, we should expect the ions 
from potassium sulphate to be atoms of potassium less a 
negative electron, and so on. We should thus expect the 
values of e/m to exhibit a regular diminution, and those of m 
a regular increase in proceeding from lithium sulphate to 
cesium sulphate. The difference of the values in these 
sequences will, of course, be greatest if the ions are the atoms 
of the elements whose compounds are used: if the ions are 
themselves of a compound nature, the differences will be not be 
so great, but will be in the same direction. Thus both the abso~ 
lute values of e/m and mand their relative values, also, should 
afford us valuable information as to the structure of these ions. 

The first experiments made were of a rather qualitative 
character. The method was adopted of comparing the de- 
flexion of the ions of the salt used with that of the platinum 
ions from the same strip and in the same magnetic field. It 
was found that the ions from sodium sulphate were deflected 
rather more, and those trom potassium sulphate rather less, 
than those from platinum. This was in accordance with 
previous expectation. A surprise was in store, however, 
when lithium sulphate was tried. It was found that on the 
first trial the ions from this substance were deflected to 
almost exactly the same extent as those from platinum but 
rather less if anything, whereas on the view expressed above 
the deHexion for the ions from lithium sulphate should have 
been very much bigger than for those from the two metals 
of higher atomic weights. Instead of this, the values for 
lithium sulphate were between those for the sulphates of | 
sodium and potassium. 


— 
—— 


Thermions emitted by the Alkali Sulphuates, 985 


Tt occurred to the author that the explanation of this 
result might be as follows: It is well known that at low 
temperatures, when the salts of the alkali metals are ionized 
by heat, the amount of ionization produced is much greater 
the higher the atomic weight of the metal used. ‘This 
tendency is well shown, for example, in experiments by 
Smithells, Dawson, and W ilson * on the conductivity im- 
parted to flames by alkali salt vapours. Now lithium 
sulphate is certain to contain the sulphates of potassium and 
sodium as impurities, and these will give rise to an amount 
of ionization more than proportionate to the amount of them 
which is present. In fact, it is quite conceivable that the 
bulk of the ionization from the lithium sulphate at first may 
be caused by the comparatively small quantities of the 
sulphates of the metals of higher atomic weight mixed 
with it. On this view, we should expect that the value of 
e/m would increase and that of m diminish with continued 
heating of the lithium sulphate, and for two reasons. In the 
first place, the decomposition and consequent removal by 
ionization is greater the greater the atomic weight, and, in 
addition, the volatility of the salts of the alkali metals in- 
creases as the atomic weight of the metal increases. Careful 
observations were therefore made from time to time on 
the deflexion, in a magnetic field, of the positive ions from 
a specimen of lithium sulphate, which was heated continuously 
for about 70 hours. It was found that the deviation of the 
ions produced by the magaetic field gradually increased with 
the lapse of time, and in fact the results were in complete 
accordance with the view that the initial values were due to 
the presence of salts of the metals of higher atomic weight. 

The results of the observations on the different salts will 
now be considered in detail in the order of the atomic weights 
of the constituent metals, commencing with lithium sulphate. 


Lirnium SULPHATE. 


The lithium sulphate used was supplied as pure by Messrs. 
Eimer & Amend, New York. As was the case with all the 
sulphates investigated, experiments were first made with the 
initial ionization from the platinum itself. The residue of 
this was then driven off by heating all night at a somewhat 
higher temperature. After the emission of positive ions from 
the platinum itself had falien to a small value, the platinum 
was taken down and a small quantity of lithium sulphate laid 


* Phil. Trans. A. vol. 193. p. 108 (1899). 
Phil. Mag. 8. 6. Vol. 20. No. 120. Dec. 1910. a 8 


986 Prof. O. W. Richardson on the Positive 


on the strip.. The latter was then heated until the salt 
melted. In this way, a uniform layer of the salt was obtained 
over the whole of the front surface of the strip. The 
apparatus was again set up and exhausted. The distribution 
of the positive “jonization on the opposite planes was now 
determined after the salt had been heated for different lengths 
of time. Throughout this investigation a difference of 
potential of 200 ‘volts was maintained between the plane 
containing the strip and that containing the slit, the strip 
being positive. In all the experiments w ara Lithia sulphate 
the transverse magnetic field was of strength 4700 lines 
per cm., and measurements were made with it first in one 
direction and then in the other. In some cases experiments 
were also made in the absence of a magnetic field. The 
distance z between the plates near the slit and the back of 
the strip was measured, by focussing with a microscope 
furnished with a micrometer arrangement for vertical dis- 
placements, from time to time during the course of the 
experiments. The apparatus was continuously exhausted by 
means of a Gaede pump, and the pressure recorded on the 
McLeod gauge was kept below ‘001 mm. In some of the 
experiments “it was as low as 10-5 mm. Provided the 
pressure is less than ‘Ol mm., its actual value does not seem 
to influence the results. 

The results of the experiments can best be exhibited by 
means of curves showing the fraction of the total number of 
ions received by the plates which pass through the slit for 
different positions of the latter. The rationale of this is 
fully explained in the first of the papers already referred to *. 
The units used are arbitrary since a capacity of ‘001 micro- 
farad was always placed in parallel with the plates, whereas 
there was no capacity except that of the electrometer and its 
connexions attached to the slit electrode ; and the capacity 
of the electrometer and its connexions was not determined. 
lt was, however, the same in all the experiments. 

The results of the experiments on the ionization from 
platinum alone are shown in fig. 1(Pl. XX.). These represent 
two successive series of measurements with the magnetic field 
in each direction. The points for the first series are shown 
thus: ©, and for the second series thus: x. Although the 
curves are not quite coincident, the displacement of the 
maximum which gives the values of e/m and m is practically 
identical for both sets. At the beginning of this experiment, 
two measurements of the value of < gave ‘480 and ‘478 cm. 


* Phil. Mag. [6] vol. xvi. p. 740 (1908). 


Thermions emitted by the Alkali Sulphates. 987 


respectively, and at the end :489 and ‘485. The mean of 
these values of z is ‘483 cm. 

The observations taken immediately after the lithium 
sulphate was placed on the strip are exhibited in fig. 2. Here 
again we have the result of two determinations taken im- 
mediately after one another, for each direction of the magnetic 
field, the first being shown thus: ©, the second thus: x. 
As before, the corresponding curves are not quite identical 
but the displacement of the maximum is the same for each 
pur. Itis slightly less than that for platinum, indicating 
that the ions emitted by fresh lithium sulphate are slightly 
heavier than those emitted by that metal. The curves are 
also rather broader than those for platinum. Immediately 
before this series of measurements was made the value of z 
was measured and found to be ‘485 cm. These curves have 
only one maximum, showing either that the ions are all of 
one kind, or at any rate that the value of e/m for the different 
kinds, if more than one, does not vary enough for them to be 
separated by the magnetic field used. 

The next curve (tig. 8) shows the effect of heating the 
lithium sulphate continuously for twelve hours. The right 
and left hand curves, points thus: @, are those obtained 
when the magnetic field is applied ; the central curve, points 
thus: x, 1s what is obtained in the absence of the magnetic 
field. We see that the curves obtained in the presence of 
the magnetic tield are now more complex than before. In- 
stead of having a single maximum we now have two maxima 
separated by a minimum. The obvious explanation of this 
is that there are now two kinds of ions present which’are 
differently deflected by the magnetic field. That this is the 
correct explanation is shown by the curve obtained in the 
absence of the magnetic field. This possesses only one 
maximum, and is in all respects similar to those obtained 
earlier, in the magnetic field. The shape of this curve 
entirely precludes the possibility that the humps are due to 
some irregularity which has developed in the distribution of 
the salt along the strip. 

The distance between the two inner humps, corresponding 
to the heavier particles, is about the same as that given by a 
new platinum wire or a fresh specimen of lithium sulphate ; 
whereas the distance between the two outer humps is very 
much greater and corresponds to ions of much smaller mass. 
The value of z in this experiment was °477 cm. 

As the heating is continued it is found that the outer hum 
grows whilst the inner hump diminishes. After 30 hours 
heating the outer humps were much bigger than the inner 


al 2 


—E——=—==——— i eEoO 


Eo EO EE EEE EEE EO OO EE EEE 


988 Prof. O. W. Richardson on the Positire 


ones. The whole pattern was also rather irregular, and part 
of this irregularity persisted in the absence of the magnetic 
field. his distortion is probably due to the salt not being so 
uniformly distributed after all this heating as it was at first. 
The appearance phertgh after 44 hours heating i is shown in 
the next curve (fig. 4). Here the main part of the curve 
consists of the two well-marked outer humps. ‘The value of 
z was not measured at this stage, but it may be taken as 
‘480 em., which is half way between the value -477 cm. after 
12 hours heating and the value °4835 cm. attained alter 
70 hours heating. 

When the heating was continued further, it was found that 
the ionization got smaller and smaller, 50 that increasing 
temperatures had to be employed in making the measure- 
ments. In addition,a small hump gradually developed on the 
inside of the outer one. This got ‘larger whilst the old one 
became smaller and smaller, After about 70 hours heatin 
there was ayain practically only one hump, and the displace- 
ment of this on reversing the magnetic tield was very little 
bigger than that given by the ions emitted from fresh plati- 
num. The curves obtained after heating for 70 hours are 
shown in the next diagram (fig. 5). The breadth and irre- 
gularity of the curves is probably due to there still being a 
small amount of the lithium salt present. Two measurements 
of zat the end of this experiment gave ‘484 and *483 em. 
respectively, the mean of which is *4835 cm. On attempting 
to carry the experiment still further it was found necessary, 
in order to get a measurable amount of ionization, to raise 
thé strip to a ‘temperature so high that it melted. 

The foregoing results can be readily explained if we assume 
that the positive ionization emitted by fresh lithium sulphate 
is due to the salts of the alkali metals of higher atomic weight 
(sodium and potassium) which it contains as impurities ; that 
the ionization which these substances produce at a low 
temperature is so great in comparison that it completely 
masks that due to the much larger quantity of lithium 
sulphate present, and that the continued heating drives off the 
sulphates of higher atomic weight, so that we then get the 
much bigger displacements due “to the lighter ions given off 
by the lithium salt itself. Finally if the heating is continued 
long enough the lithium salt is driven off. We then get, at 
a much higher temperature, a small quantity of the ionization 
characteristic of the platinum itself. 

The evidence in favour of this view has so far been of a 
rather qualitative nature. It receives additional support 
when the values of e/m and m deduced from the above 


Thermions emilted by the Alkali Sulphates. 989 


experiments are considered. The numerical results are 
exhibited together in the following table :— 


Inthium Sulphate—Positive Ions. 


Seria, Time | z Vv H £ | _ejm m 
heated | (ems.)| (volts). | (lines |(1=-063/(E.M. | (H=1). 
(hours). percm,*) ‘en ) units). 
eas: 0 |-483 | 200 | 4700 | -99 | 297 | 325 
SO, ...).. 0 |.485 | .200.-|. 4700 ‘95 269 | 35:9 
pe) oe VDI 47. /9t 200% hy A700 ‘8D 231 | 41-8* 
Pa TI ATR |e 20004 4F09 2h B35 L7G0%: ol LESTE 
Peel sarge a00e | 4700 8 RBSS 4} 1785 «| SBT" 
eee) o2 | 48h) | 260, f° 4700") oe | 1300") 743) | 
ok Si 70 | 4885} 200 | 4700 | 1:25 470 | 206 


* The value calculated from the two inside humps. 
Tt The value calculated from the two outside humps. 


The first column gives the material experimented with, the 
second shows the time the lithium sulphate had been heated, 
the third gives the distance in ems. between the back of the 
strip and the plate, the sixth gives half the displacement of 
the maximum produced by reversing the magnetic field given 
in the fifth column, whilst the meaning of the numbers in 
the remaining volumns are obvious. The values of .m repre- 
sent the ratio of the mass of the ions to that of an atom of 
hydrogen on the assumption that the ions carry the same 
charge as that carried by an atom of hydrogen in electrolysis. 
The value of e/m for hydrogen was taken to be 9°66 x 10° 
E.M. units. 

The value of m for the platinum ions (32°5) is not very 
different from the mean (25:7) of the values given by 
Richardson and Hulbirt*, and is within the limits of the 
range of values fonnd by them. On the view that the 
positive ions emitted by hot metals are due to the presence 
of alkaline impurities, this value would correspond to that 
from a mixture of potassium (atomic weight 39°15) and 
sidium (atomic weight 23°05). This is on the assumption 
that the ions are positively charged atoms of the metals in’ 
question. We shall see that this is substantiated in so far 
that the positive ions emitted by the sulphates of the alkali 
metals are atoms of the metals they contain. 

The value of m for the ions from fresh lithium sulphate 
(35°9) is very near the atomic weight of potassium, and is in 


Ae GA Gi = 


—_—_ SSS ee ——_— ll Te  -  C 


: 
: 
a 
| 


990 Prof. O. W. Richardson on the Positive 


agreement with the view that the initial ionization is due 
principally to the admixture of saits of this element, perhaps 
with some sodium in addition. ‘The value 41°8, for the inner 
of the humps which developed after the twelve hours heating, 
again is of approximately the same magnitude, pointing to 
the presence in smaller relative amount of the same impurity. 
The value (5°5), calculated from the two outer humps, agrees 
satisfactorily with the atomic weight of lithium (7:05). A 
still better agreement is obtained after the inner humps have 
disappeared, and we obtain the value of m which presumably 
corresponds to the pure lithium salt. The value after 44 
hours heating was found to be 5:57, and after 52 hours 
heating 7°43. The mean of these values is 6°5 instead of 7:05. 
The value (20°6) obtained after 70 hours heating is what we 
should expect if it were due to the platinum ions, for which 
m was found to be 32°5, mixed with a certain amount of the 
more deviable ions from the lithium salt. Thus the view we 
have taken of these effects gives a satisfactory account of the 
phenomena exhibited by lithium sulphate. The position is 
still further strengthened when the salts of the other alkali 
metals are examined. 


SopItM SULPHATE. 


The specimens of sodium and potassium sulphates used 
were presented to the author several years ago by Professor 
H. A. Wilson. They had been speciaily prepared for use in 
obtaining standard temperatures from their melting-point, 
and are believed to be very pure. At any rate, the sodium 
sulphate is not likely to contain anything with more capacity 
for emitting ions than itself, with the possible exception of 
traces of potassium sulphate. We should therefore expect 
that sodium sulphate would behave quite differently from 
lithium sulphate. The deflexion of the maximum by the 
magnetic field ought to be quite constant, except in so far as 
any trace of potassium sulpbate present might make the 
deflexion a little smaller at first than later. ‘ihere ought to 
be no development of widely divergent humps as in the case 
of the lithium sulphate. As a matter of fact this is exactly 


“what was observed. 


As before, two sets of curves for the initial ionization 
from the platinum strip (a fresh one) were taken. The 
distance between the maxima for one set of curves was 1°96 
turns of the screw and for the other 2°04 turns, the mean 
being 2°00 turns. The value of z was‘492cm. The strength 
H ot the magnetic field in this set of experiments was 
4650 lines per cm. Theabove numbers give m=33'7, which 


Thermions emitted by the Alkali Sulphates. 991 


is practically identical with the value found from the specimen 
of platinum used in the experiments on lithium sulphate. 

In this set of experiments it was not possible to make 
measurements on the ionization from sodium sulphate until 
the heating had gone on for about two hours. Three sets of 
observations were taken, after heating for two hours, eight 
hours, and twenty-four hours respectively: The results are 
shown in fig. 6. In this experiment the field strengths &e. 
had the following values:—V=200 volts, H=4650 lines 
Ber em.*,-2=°497 cm, 

It will be seen that the maxima are deflected to nearly 
the same extent in every case. However the first curves, 
points thus: x, obtained after two hours heating, give a 
slivhtly smaller deviation (2°3 turns) than the others and 
lead to the value m=26°6. The next, points thus: x, after 
eight hours heating, give a deviation of 2°45, corresponding 
to m=23°4. After heating twenty-four hours, points thus : 
® and |x|, the deviation is practically unaltered at 2°5 turns, 
corresponding to m=22°5. ‘The effect of continued heating 
on the sodium sulphate is clearly very different from that of 
lithium sulphate. 

The very small size of the humps obtained after heating 
the sodium sulphate for twenty-four hours seemed suspicious, 
and was investigated further. It was found that these two 
humps represented only an isolated portion of the curve, 
which really extended over a much greater length in the 
direction of « than that shown in the figure. Detailed ob- 
servations taken after thirty hours heating showed that a 
measurable fraction of the ionization passed through the slit 
all the way from «=10 turns to z=11 turns. There were 
several humps in the curve, which resembled a panoramic 
view of a range of mountains. There was, however, no 
evidence of any splitting up of the ions into groups charac- 
terized by different values of e/m. The patterns were displaced 
asa whole by the magnetic field, corresponding points being 
displaced about equal distances, and the form of the curve 
was the same in the absence of the magnetic field as with it. 
It was quite different from the behaviour of lithium sulphate 
shown in fig.3. The peculiarities observed could be explained 
if the sodium sulphate after continued heating tended to 
collect in lumps at the edges of the strip where the electric 
field is irregular. When the apparatus was taken down, the 
salt which was left was too small in amount to be visible. 
It was observed, however, that the strip had become contorted 
somewhat, which would also expiain the peculiar effect 
encountered. 

Another set of observations made on sodium sulphate 


i 


S92 Prof. O. W. Richardson on the Positive 


confirms the preceding results and brings out one or two new 
points as well. This set is represented in fig. 7. The curves 
at the top (fig. 7, A), points thus: x, exhibit the effect 
of heating the platinum by itself. In these experiments 
Ve=Z00 volts, H=4700 lines per cm.”, and z=*556 cm. 
The displacement 2°85 turns for the platinum ions leads to 
the values e/m=349 and m= 27'7, in agreement with those 
found previously. 

Three curves are given, fig. 7, B, representing observations 
on sodium sulphate. Those with points thus: ©, were 
obtained when the salt was first heated. It will be noticed 


that they are broader than and not so high as the later ones. - 


The displacement of the maximum, 2°95 turns, corresponds to 
m=25°9. It is greater than that for the platinum i ions, and 
less than that, 3°3 turns, obtained after the sodium sulphate 
had been heated longer. The two remaining curves were 
taken in succession after the salt had been heated for about 
fourteen hours. The curves with points thus: XX, were 
taken first, those with points @ and |x/ later. The displace- 
ment of the maximum is the same in both cases, and the 
character of tle curves is very much the same. The value 
of m ealeulated from the displaceme nts is 22°). 

The greater value of m and the greater breadth of the 
curves obtained on first heating is readily explained if we 
suppose that the sodium sulphate coniains a small quantity 
of potassium sulphate which would be much more effectively 
ionized, at the lower temperature at which the observations 
pranienone: 

The results of the measurements with the sodium sulphate 
are gathered together for reterence in the following table. The 
units &c. are the same as in the table for lithium sulphate. 


Sodium Sulphate—Positive Ions. 
sceeianeesaeoe Eni 


: a H oe e/m 

Substance.| jeated. | (ions), (volts). esta ic crag leat (H=1), 
Platinum .| 0 492 200 4650 1-00 287 33°7 
Na,SO,...) 2 ‘497 | 200 | 4650 | 1-15: | 368°) ae 
Na,SO,...| 8 ‘497 | 200 | 4650 | 1:295| 418 | 284 
Na,SO, ...| 24 ‘497 | 200 | 4650 | 1:25 | 430 | 2295 
Platinum .| 0 556 200 4700 1425 349 255 
Na,SO, ...| 0 556 | 200 | 4700 1475 | 373 | 259 
| -Na,SO, ...} 13 556 | 20 700 160 | 439 | 220 
| Na,€O, ...] 15 556. | 260 |. 4700 | 1-600) a 22-0 


Thermions enitled by the Alkali Sulphates. 993 


PoTassiIUuM SULPHATE. 


Several sets of observations were made on the ionization 
from the fresh platinum strip which was used in this experi- 
ment. In these experiments the values of the field &c. were 
M200: volis, H=4650 lines per cm.7, and c=*501 ecm. 
Immediately after the heating of the strip was commenced 
the displicement 2x of the maximum was found to be 2°42 
turns, giving e/m=390 and m=24°8. After heating for 
about one hour 2.c had become 2°28 turns, giving e/m=346 
aud m=27°9. After six hours heating it was found Mee 
2x= 2°00 turns, e/m=266, and m=36'3. 

The gradual increase of m as the heating is conta 1s 
in agr eement with the author’s previous experiments on 
platinum *. It would seem to indicate that the source of the 
positive ionization of lower atomic weight from platinum is 
more easily driven off by heating than nee of higher atomic 
weight. The results are in agreement with the view that 
the ions of lower atomic wel ight are sodium atoms, but as the 
substance of higher atomic weight is less easily driven off by 
heat, it would appear that some impurity other than potassium 
or its salts has to be looked for. So far the writer has not 
been able to observe the development of humps in the curves 
for the ionization from platinum similar to those frem lithium 
sulphate. This would seem to indicate, if the observed 
change in the nature of the ions is real, that it takes place 
very ‘oradually, as there is no evidence of the simultaneous 
emission of two groups of ions with widely separated values 
of e/m. The experiments on the platinum ionization were 
not continued further, as the ionization at the temperature at 
which the experiments were being carried out became very 
small at the end of the last experiment. 

The curves given by potassium sulphate itself were very 
simple in character. They consisted of just a single hump, 
and remained almost unchanged during sixty hours heating. 
At the same time the displacement of the maximum was 
constant within the limits of experimental error. After 
about eighteen hours heating of the salt, however, the curves 
broadened very considerably, but this may have heen due to 
some temporary contortion of the strip as it disappeared 
Jater, and all the time afterwards the curves were sim] le 
as at first and gave the original value of e/m. 

The results of the first few heatings of the salt are exhibited 
in fig. 8. The left-hand curve with points thus :—x, was 
taken as soon as the heating commenced. ‘lhe magnetic 


* Cf. Phil. Mag. [6] vol. xvi. p. 769 (1908). 


954 Prof. O. W. Richardson on the Positive 


field was then reversed, and the curve with points thus :—x, 
taken. The field was again changed back to its original 
value and the position of the first maximum redetermined. 
The points are shown thus:— x]. The position of the first 
maximum was found to be unchanged within the limits of 
experimental error. 

The curves with broken lines, points @, represent similar 
observations taken after six hours heating. Although the 
positions are displaced a little, the displacement of the maxi- 
mum due to the field is the same as in the preceding case. 
In the experiments represented in this diagram the values of 
the numerical quantities. were:—V=200 volts, H=4600 
lines per cm., -='501 cm., and 27=1:98 turns. These 
values, which are the same for both sets of observations, give 
e/m=261 and m=37°0. 

Similar sets of observations were taken after 18, 24, 36, 
42, and 60 hours heating. As, except for the temporary 
broadening mentioned above, which was observed after 
eighteen hours heating, all the curves were similar in 
character, it will be sufficient to give the last one of them. 
This is shown in fig 9, and represents the results obtained 
after sixty hours heating. The data pertaining to this curve, 
as well as the other observations mentioned, are given in 
the following table :— 


Potassium Sulphate—Positive Ions. 


| | | ae 


Time | r wl) be H | Su | elm ra 
jana oes bea (volts). ene i Saat ee | (Hah, 
eM es A) PE A OS wey) 
Platinum, 0 | “501 200 | 4630 | 1:21 390 | 24:8 
a Laie | 501 200 | 4650 | 1:14 | 346 27°9 
n | 6 | ‘DUL 200 | 4650 1-00 | 266 36:3 
Oe 4+) oA 13501 200 4650 OO" "| 2a 37 
Ae adr (of ebots 200 4650 | 99 | 261 | 87 
; if 24 486 | 200 | 4650 | 98 261 | 37 
cpa Nplate: BR = 486 200 | 4650 ‘95 | 272 Sao: 
mit wet 42. 486 | 200. 7) 4Gap ‘94 266 36°3 
IAS aE le D2 a "5, | 4650 ‘94 | 966 | 


| | 
; 
} 

| 


| 
| 
| 
36:3 | 
Mean value of m for K,SO, = 36% | 


It will be seen that the different values of ¢/m and m for 


Thermions emitted by the Alkali Sulphates. 995 


the positive ions from potassium sulphate given by this set 
of observations are all very near one another and quite close 
to the value for the atom of potassium. The atomic weight 
of this metal is 39°15. 

Another set of observations on potassium sulphate gave 
for the initial ionization from platinum 22=1°88 turns. 
The corresponding quantity immediately after heating potas- 
sium sulphate was found to be 1°80 turns. The value of z 
for this experiment was not measured, so that the values of 
e/nvand m could not be calculated. The relative values for 
platinum and potassium sulphate, however, confirm those 
obtained in the more complete series of experiments just 


described. 


RUBIDIUM SULPHATE. 


The ionization obtained from the platinum strip used in 
this experiment gave a high value of m, probably owing to 
the strip having been heated for some time before the 
measurements were made. The data for this part of the 
experiment are:—V= 200 volts, H=4700 lines per cm.’, 
— oi2 em., and 27 = 2-00 turns. These values give 
e/n=240 E.M. units and m=40°3. 

The results of the observations on rubidium sulphate are 
exhibited in fig. 10 (Pl. XX.). These were taken shortly after 
commencing to heat the salt, but the exact interval of time 
was not recorded. The left-hand curve with points thus :— x , 
was taken first. The magnetic field was then reversed, and 
the right-hand curve with points thus:— x, obtained. The 
original direction of the magnetic field was now restored, 
and the lett-hand points @ were obtained. The field was 
again reversed, giving the right-hand points, ©. The dis- 
placement between the first two maxima is 1-24 turns, that 
between the two second 1°30 turns. The mean is 1°27 turns 
The other data for this experiment are V=3200 oli! 
H=4700 lines per cm.?, and z=‘506 cm. These’ give 
e/m=101 and m=96. The atomic weight of rubidium 
= 00°): 

The experiments with rubidium sulphate were not con- 
tinued further, as it was thought at the time that this was a 
satisfactory agreement. It may be, however, that the rubi- 
dium sulvhate contains some cesium as an impurity, and 
that further heating would have increased the value of e/m. 
Jt is worthy of note in this connexion that rubidium is the 
only element so far for which the value of m is distinctly 
above the atomic weight, the other cases tending to fall 


996 Prof. O. W. Richardson on the Positive 


below it. However, the difference is not greater than the 
error of observation might well account for. 

The specimen of rubidium sulphate used was purchased 
from Messrs. Eimer and Amend. It was not stated to be 


especially pure. 


CzsSIUuM SULPHATE. 


Two sets of observations were made with the ionization 
from the fresh platinum strip used in this experiment, one 
immediately after the heating was started and the other : 
few hours later. The first led to the values 27=2-05 turns, 
e/m= 344, m= 28'2, and the second, 2v7= 1°95 turns, e/m=295, 
m=32"7. . 

This increase with lapse of time in the value of m for 
the ions from platinum is in agreement with that already 
discussed under the heading of potassium sulphate. 

The following data were the same for all the experiments 
in this series, whether with platinum alone or with ca#sium 
sulphate :—V= 200 volts, H = 4700 lines per cm.?, and 
2=474 cm. 

The results of the observations on ceesium sulphate are all 
shown in fig. 11. ‘The first two curves, points:— x, were 
taken immediately after the heating had started. It will be 
noticed that these are distinctly broader than the later 
curves, and the displacement of the maximum was greater. 
The displacement in this case was 2a=1'125 turns, giving 
e/m=101°8, m=!5. 

The curves with points @ were obtained after eighteen 
hours heating. The displacement now is 27=°'85 turns, 
giving e/m =5%1 and m=163. The remaining curves, 
points @ and |x|, represent the results after twenty-three 
hours heating. The displacement is again 27='89d turns, 
giving the same values of ¢/m and m as the last. 

It may be that the greater breadth of the curves and the 
greater displacement of the maxima obtained initially repre- 
sents a real difference in the ionization. It is, however, 
difficult to be certain about the ma‘ter, as the displacements 
with these heavy ions are small, and the probable experi- 
mental error is correspondingly great. The mean of the 
three measurements gives m=140. The atomic weight of 
ceesium is given as 133°9 

The specimen of ceesium sulphate used was prepared by 
the action of pure sulphuric acid on the chloride. The 
latter was bought from Messrs. Eimer and Amend, with no 
specification as to its purity. 


7 


Thermions emitted by the Alkali Sulphates. 997 


General Considerations. 


The values of e/m and m which, according to the view of 
these phenomena developed in the present paper, are cha- 
racteristic of the positive ions emitted by the pure sulphate of 
the alkali metals, are collected together in the following 
tible. The values which are believed to be due to the 
presence of impurities in the material are omitted from the 
table. 


Time e/in mys | Average Atomic 
Substance. heated (E.M. BUN) pepe) value of | weight of | 
(hours). units). mM. metal. 
| | 
DePSOP sy. 12 1760 55 | | 705 
MSO), | 44 1735 Boe iets 7-05 
6 ee wba 1300 743 | 62 703 | 
Na SO 2.4... | 8 413 23-4 | 23°05 | 
PO lt 430 a5 23:05 | 
RESO 2). pees 439. Jo (220 || 23-05 
ROO «22s. Beil pina do UE Nec BRO. | BRS 23-05 
ESO. 5:.08051| 0 261 SON s | 8915 | 
BSG ic... 6 OG I) «| Se se 39°15 
WO, ess... 24 Lh a 39-15 
BNO 2543... 36 272 | Seo | 89°15 
Oe... 42 CTE SES al 39°15 
REG 60 26 | 363 | 3865 39-15 
i HOES 4 8Gius book IG 85°5 
CRS utes. <. 0 1018 | 95 | 132-9 
Ge sOpne:..v.. HS pildissebas by 163 ih 132-9 
Ca cde ue 23 591 | 163 140 132-9 


The numbers in the last three columns of the table show 
conclusively that the positive tons emitted by the sulphates of 
the alkali metals are atoms of the constituent metal. The 
range of e/m, m, and of the atomic weight for the different 
metals is so sreat that, although the accuracy of the measure- 
ments is not, perhaps, all that might be desired, particularly 
in the case of the elements of higher atomic weight, never- 
theless the general agreement of m over so wide a range 
is such as to leave no room for escape from the above 
conclusion. 

It also follows that the positive ions emitted by the sulphates 


998 Prof. O. W. Richardson on the Positive 


of the alnxavt metals carry the same charge as that carried by 
an atom of hydrogen in electrolysis, since this assumption has 
been made in calculating the values of m and leads to con- 
sistent results. We therefore conclude that the positive 
ions emitted by these substances when heated have the same 
structure as that which is generally attributed to the nuclei 
of the positive ions to which the same substances give rise 
in solution. In other words, they consist of’ one atom of the 
basic metal which has lost a negative electron. 

The similarity between the thermions and the electrolytic 
ions appears to end with the positive ions. So far the only 
negative thermions which have been detected have a very 
large value of e/m and are negative electrons. There is no 
evidence of the existence of negative thermions similar in 
structure to those which occur in solution. 

The present investigation does not throw any definite 
light on the chemistry of the processes involved. Since the 
positive ions are atoms of the metal, it is clear that they 
must get free from the rest of the molecule somehow. It is 
possible that the first decomposition consists of a splitting up 
of the sulphate into positively charged atoms of the metal 
and a negatively charged SO, group, as is the case in 
solution. ‘This would then presumably be followed by the 
breaking up of the negatively. charged group into its chemical 
constituents together with a negative electron. The results 
of the experiments neither affirm nor deny this view. It 
seems possible that valuable evidence in this connexion 
might be obtained by looking for a relation between the 
amount of positive and of negative ionization produced by 
heated salts. Experiments with this end in view are being 
instituted. In any event, the present investigation shows 
that the expulsion of a positively charged atom of the metal 
is a salient feature of the ionization of heated salts. 

The balance of evidence at present is not in favour of the 
view that the positive ions produced by salt vapours in 
flames at a high temperature are identical with those here 
investigated. Their velocities have been measured hy dif- 
ferent experimenters, and there is a general agreement that 
the positive ions produced by the salts of all the alkali 
metals have the same velocity under unit field. This result 
would not be expected if the ions were atoms of the metals. 
The absolute values of this common velocity given by different 
observers are rather divergent. For instance, H. A. Wilson 
finds 62 cm. per sec., Marx finds 200 em. per sec., and Moreau 
80 cm. per sec. Nevertheless, all these values are higher 
than what would be expected if the ions were atoms of the 


Thermions emitted by the Alkali Sulphates. 999 


metal, particularly in the case of the elements of higher atomic 
weight. They are, however, in very good agreement with 
the value calculated from the formula u = : = x where w is 
the velocity under unit field, e the charge, m the mass, % the 
mean free path, and v the m2an velocity of agitation (ok the 
ions, on the hypothesis that thev have the same structure as 
those discovered by Sir J.J. a * in the canal rays. 
Moreover, the ratio of the velocity of the negative to that “of 
the positive ions in flames at high eoipce tae: is in good 
agreement with this view. 

“Outside of solutions the number of cases of positive ions 
of which the structure has been definitely determined is not 
very large. The only ones which occur to the author are 
the « particle, which Rutherford showed to be an atom of 
helium carrying twice the electronic charge, and the canal 
rays in different gases, for which the mass of the carriers 
has been investigated by Wien, Stark, and Thomson. With 
the exception, possibly, of the particles having a high value 
of e/m, isolated by Thomson from the canal rays, all these 
researches agree with the results of the present investigation 
in making the primitive positive ion an atom of some known 
chemical element which has lost one or more negative 
electrons. 


In concluding I wish to thank Mr. Irving B. Crandall, 
A.B., graduate student in physics, and my assistant, Mr. Cor- 
nelis Bol, for their help in taking the observations. 


Palmer Physical Laboratory, 
Princeton, N. J. 


CXI. The Positive Thermions emitted by the Salts of the 
Alkali Metals. By O. W. Ricwarpson, J.A., D.Sce., 
Professor of Physics, Princeton University +. 

1" the preceding paper the author has shown that the 

positive ions emitted by the sulphates of the different 

alkali metals are atoms of the constituent metal carrying a 

single electronic charge. The sulphates were used in the 

investigation because they are readily obtainablein a state of 
reasonable purity and are comparatively non-volatile. There 
is no reason to expect that the sulphates are exceptional in 
regard to the positive ions they emit. We should, therefore, 


*® Phil. Mag. [6] vol. xvi. p. 657. (1908). 
+ Communicated by the aaihoe 


1000 Positive Thermions emitted by Salts of Alkali Metals. 


expect that all the salts of a given alkali metal would emit 
positive ions having a value of e/m which is the same for all 
of them and is equal to the value of that quantity for the 
salts of the corresponding metal in electrolysis. 

Experiments to test this point were made with sodium 
fluoride and sodium iodide as well as sodium sulphate. 
Sodium salts were chosen as it is desirable to use an element 
of low atomie weight ; since the influence of the non-metallic 
part of the compound will then be relatively greater. 
Lithium was rejected on the ground that its salts are liable 
to contain those of the elements of higher atomic weight as 
impurities, and the ionization from these is apt to mask that 
from the lithium salt. The fluoride and iodide were chosen 
as they ditfer widely in their molecular weight from each 
other and in chemical constitution from the sulphate. 

Measurements of the e/m for the positive ions from sodium 
fluoride were made after two hours, four and a half hours, 
sixteen hours, and thirty-six hours heating. Similar measure- 
ments with sodium iodide were made after two hours, six 
hours, and eighteen hours. In neither case did any of the 
values obtained differ from the electrolytic value for sodium 
salts by more than five per cent., which is about the order of 
accuracy claimed for the observations. 

We therefore conclude that the positive ions emitted by 
the salts of the alkali metals depend only on the nature of 
the constituent metal and are, in fact, atoms of the metal 
which have lost a negative electron. 

Since the alkali metals are monovalent we should expect, 
on chemical grounds, that their ions would contain only one 
unit of electronic charge. In the case of the metals of 
higher valency we stand a better chance of getting ions 
which carry more than one unit of charge. Although, in 
any case, multiply charged ions will have to contend with a 
much stronger tendency to recombination, and will, on that 
account, be less likely to be liberated than singly charged 
ions. For these reasons the nature of the positive ions 
liberated by the salts of the other metals when they are 
heated is of great interest, and expcriments on them are now 
being made by the same method. 


Palmer Physical Laboratory, 
Princeton, N.J. 


[1008 J 


CXII. The Problem of the Whispering Gallery. 
By Lord Rayurter, OM. PRS.* 


| tee phenomena of the whispering gallery, of w hich 

there isa good and accessible example in St. Paul’s 
cathedral, indicate that sonorous vibrations have a tendency 
to cling i a concave surface. They may be reproduced 
upon a moderate scale by the use of sounds of very high 
pitch (wave-length=2 cm.), such as aré excited by a bird- 
call, the percipient being a high pressure sensitive flame ft. 
Especially remarkable is the narrowness of the obstacle, held 
close to the concave surface, which is competent to intercept 
most of the effect. 

The explanation is not difficult to understand in a general 
wav, and in ‘Theory of Sound,’ § 287, I have given a cal- 
culation based upon the methods employed in geometrical 
opties. I have often wished to illustrate the matter further 
on distinctively wave principles, but only recently have re- 
cognized that most of what I sought lay as it were under my 
nose. The mathematical solution in question is well known 
arid very simple 1 in form, although the reduction to numbers, 
in the special cir cumstances, , presents certain difficulties. 

Consider the expression in plane polar coordinates (1, @) 


eo Gh) Cos (kat nO her, 2. (1) 


applicable to sound in two dimensions, Ww denoting velocity- 
potential ; or again to the transverse vibrations of a stretched 
membrane, in which case wy represents the displacement at 
any pointt. Here a denotes the velocity of propagation, 
| 2m], where A is the wave-length of straight waves of 
the given frequency, n is any integer, and J, is the Bessel’s 
function usually so denoted. The waves travel cireum- 
ferentially, everything being reproduced when @ and ¢ 
receive suitable proportional increments. For the present 
purpose we suppose that there are a large number of waves 
round the circumference, so that n is oreat. 

As a function of r, vv is proportional to J, (kr). “When 
z is great enough, J, (z),as we know, becomes oscillatory 
and admits of an infinite number of roots. In the case of 
the membrane held at the boundary any one of these roots 
might be taken as the value of &R, where R is the radius of 
the boundary. But for our purpose we suppose that cR is 

* Communicated by the Author. 


+ Proce. Roy. Inst. Jan. 15, 1904. 
t ‘ Theory of Sound,’ §§ 201, 339. 


Phil. Mag.8. 6. Vol. 20. No. 120, Dec. 1910. 3U 


1002 Lord Rayleigh on the 


the jirst or lowest root (after zero) which we may call 2. 
In this case J, (z) remains throughout of one sign. For the 
aerial vibrations, in which we are especially interested, the 
boundary condition, representing that r=R behaves as a 
fixed wall, is that J,’(AR)=0. We will suppose that * and 
R are so related that /R is equal to the first root (z,') of this 
equation. The character of the vibrations as a function of 
y thus depends upon that of J, (2), where » is very large 
and z less than <, or z,.. And we know that in general, 
n being integral, 
te teak 
Jg{¢) == cos (csinwe—nw)do.. . . (2) 


Moreover, the well known series in ascending powers of ¢ 
shows that in the neighbourhood of the origin J» (¢) 18 very 
small, the lowest power occurring being <”. 

The tendency, when n is moderately high, may be recog- 
nized in Meissel’s tables*, from which the following is 
extracted :— 


Z. Jis (2). Jo, (2). Z. ' Jis (2). |. . liga te 

24. —0-00381 +0°2264 16 | +0°:0668 | +0°0079 

28 +0340 (2381 | 15 | 00846 00031 

22 | 071549 O2105 | 14 | 00158 U'OOLO 

21 | 02316 1621 1 13 | 00063 00003 

20 | 02511 01106 || 12 0:0022 | O00 
| 19 (2235 O-U675 11 | 00006 | 00000 
, as O:1706 0 0369 10 | 00002 


| 3a 0'1138 O-UL80 9 | 0:0000 


From the second column we see that the first root of 
Ji3 (2)=0 oceurs when -=23'3. The functionis a maximum 
in the neighbourhood of ¢=20, and sinks to insignificance 
when z is less than 14, being thus in a physical sense limited 
to a somewhat narrow range within z=23°3. 

The above applies to the membrane problem, In the case 
of aerial waves the third column shows that Jy; (z) is a 
maximum when z=23°3, so that Jo,’ (23°3)=0. ‘This then 
is the value of KR, or z,’.. It appears that the important 
part of the range is from 23°5 to about 16. 

The course of the function J, (z) when n and z are both 
large and nearly equal has recently been discussed by Dr. 
Nicholson+. Under these circumstances the important part 


* Gray and Matthews’ Bessel’s Functions, 
+ Phil, Mag. xvi. p. 271 (1908) ; xviii. p. 6 (1909). 


Problem of the Whispering Gallery. 1003 


of (2) evidently corresponds to small values of w. If z=n 
nd we may write ultimately 


i=) 


He ; fl i 
eC) = — cos n(wa— sin w)\do= — cos 2(@— sin w)dw 
ag e 0 TT e’ 0 


@m 
ya na” Ne ie i 
- cos—— dw = = cos 2? da 
7 No 6 T\n 
SME kOe WT Nig oe kh a ae an) 


one of Nicholson’s results. 
In like manner when n—z, though not zero, is relatively 


a (1) may be made to depend upon Airy’s integral. 


11 Mi<2} 
(2) == = ( cos {(n—s) otlea’tdw. . . (A) 
“0 
In the second of the papers above cited Nicholson tabulates 
Jn (¢) against 2°1123 (n—z)/:*. It thence appears that 
4955 4 


i) eT oc Maen) 


2°1123 
The maximum (about 0°67) occurs when 
Boe, ti Dre a va ea CG) 
and the function sinks to insignificance (0°01) when 
A i ae as a pC S 


Thus in the membrane problem the practical range is only 
about 2°7 ns. 
In like manner 


BUSS) were 1 2 1 
24 = rn? = | ° “~ / 9 >3 0 


so that in the aerial ne the practical range given by 


(7) and (8) is about 2°1n*. 

To take an example in the latter case, let n=1000, repre- 
senting approximately the radius of the reflecting circle. 
The vibrations expressed by (1) are practically limited to an 
annulus of width 20, or one fiftieth part only of the radius. 
With greater values of n the concentration in the imme- 
diate neighbourhood of the circumference is still further 
increased. 

It will be admitted that this example fully illustrates the 
observed phenomena, and that the clinging of vibrations to 
the immediate neighbourhood of a concave reflecting wall 
may become exceedingly pronounced. 


aU 2 


1004 The Problem of the Whispering Gallery. 


Another example might be taken from the vibrations of 
air within a spherical cavity. In the usual notation for - 
polar coordinates (7, 0, ¢) we have as a possible velocity- 
potential yr=(kr) ey (kr) sin” @ cos (kat—ng), and the 


j discussion proceeds as before. 

i A EEE OOOO oP 

So tar as I have seen, the ultimate form of J, (z) when n 
} is very great and 2 a moderate multiple of x has not been 
: considered. Though unrelated to the main subject of this 
note, I may perhaps briefly indicate it. 

| The form of (2) suggests the application of the method 
} employed by Kelvin in “dealing with the problem of water 
i waves due to a limited initial disturbance. Reference may 


also be made to a recent paper of my own*. 

When n and ¢ are great the only important part of the- 
range of integration in (2) is the neighbourhood of the place 
or places, where zsin w—no is stationary with respect to w 
These are to be found where 


COS @ ‘= njz,\. se 


from which we may infer that when z is decidedly less than 
», the total vilue of the integral is small, as we have already 
seen to be the case. When z>n, | is real, and according 
to (9) would admit of an infinite series of values. Only one, 
however, of these comes into consideration, since the actual 
range of integration is from 0 to 7. We suppose that 2 is 
sO much greater than » that w, has a sensible value. 

The application of Kelvin’s method gives at once 


Se 2\cos {z sin w,—nw, —}7} 
T)=a/ ) isin o,} 


We may test this by applying it to the familiar case where 
zis so much greater than as to make o,=47. We find 


Ji a=, /(=) .cos {z—4na—Ar},. « (1) 


the well known form. 
As an example of (10), 


J4(2n) ener cos {(\/3Jxr) n—Jm}. (12) 


Although in (2) 2 is limited to be integral, it is not difficult 
to recognize that results such as (3), (5), (12), applicable to 
large values of n, are free from this restriction. 


* Phil. Mag. xviii. p. 1, immediately preceding Nicholson’s paper — 
just quoted. . 


f 1005 J 


CXII. On Magnetic Hysteresis. 
To the Editors of the Philosophical Magazine. 


GENTLEMEN,— 


HE interesting article in your September issue in which 
Professor 8. P. Thompson applies the Fourier analysis 
to Hysteresis Curves in the magnetization of iron closes 
with some observations regarding the nature and effects of 
hysteresis to which I think exception must be taken. He 
remarks (p. 436): ‘ Hysteresis is commonly regarded as an 
irreversible process and as such involving a degradation of 
energy into heat. But in view of the present analysis of the 
hysteresis loop it is mecessary to revise this opinion.” 
I cannot see why. Hysteresis in the magnetization of iron 
means a lagging of the magnetism behind the magnetizing 
force when the magnetizing force is made to suffer any 
change. This lagging causes each stage of the process to be 
irreversible and involves a degiadation of energy into heat. 
The area enclosed by the curve of magnetism and magne- 
tizing force measures the energy so dissipated in a cyclic 
process. These fundamental facts are in no way affected by 
the analysis, and it is not apparent that any revision of ideas 
is required. Neither can one accept the statement that 
“the energy waste does not involve or produce any pheno- 
menon of lag.” It is, on the contrary, the lag of magnetism 
behind magnetizing force that involves waste of energy. 
Professor Thompson seems to have in mind, when he 
speaks of lag, the lag of current behind impressed electro- 
motive force in a circuit such as that of a choking coil, and 
he is at pains to point out that “* hysteresis does not cause 
any lag in the current.” But to suggest that it might would 
be to put the cart before the horse. It is the current that 
represents the magnetizing force, and the lag which hysteresis 
involves is the lag of something else (namely the magnetism) 
behind the current, not any lag of the current behind some- 
thing else. 
Yours faithfully, 


J. A. Hwine. 
Froghole, Edenbridge, Kent, 
5 Nov., 1910. 


f 1006 ] 


CXIV. Notices respecting New Books. 

Introduction to Physical Chemistry. By Professor Harry C. Jonus. 
New York: The MacMillan Company, 1910. $1.60 net. 
TPXHE vitality of Physical Chemistry is exhibited as much in the 

number of text-books written upon it as in the rapid progress 
which is being made in this branch of science itself. The appear- 
ance of still another will doubtless be justified, and its merits are 
such that it will probably prove a strong rival to those already in 
the field. 

Let it be said, in the first place, that the style in which the 
book is written is an excellent one. The statements are easy to 
follow ; and considering that only a rudimentary account of the 
subject-matter is intended, they are as complete as could be 
desired. A doubt will arise in the minds of some readers whether 
it is well to attempt to give something about everything rather 
than to concentrate attention in a wore thorough manner upon a 
few things. The writer follows the former course ; the result is 
a very complete elementary resumé of the principal facts, while 
the explanations of these facts are given in a more imperfect way 
when given at all. Professor Jones is obviously writing for the 
man who has no mathematical knowledge, and difficulties fre- 
quently occur under these circumstances in giving a satisfactory 
explanation. We think that he succeeds on the whole; but there 
are many places where considerable improvement could be effected. 
A notable case occurs on p. 29, where he attempts to (but scarcely 
succeeds in) giving ap explanation of the additional terms in 
Van der Waals’ equation for a gas while he has not previously 
explained the occurrence of the terms in the equation of a perfect 
gas itself. 

Leaving such points of mere exposition, we turn to the discus- 
sion of the problem (on p. 98) of the abnormal lowering of freezing- 
point of strong solutions of calcium chloride and aluminium 
chloride. The writer is correct, we think, in considering this to 
be a consequence of hydration of the dissolved salt. But we do 
not think the reasons for his decision to be very illuminative. 
It is not so much a question of there being less free water owing 
to the hydration ; for, since the total volume is not thereby much 
changed, we would not expect as a consequence much influence 
(on the gas-theory) upon the osmotic pressure. The important 
fact to emphasize is that the size of the molecule is thereby 
increased, and just as in the case of a gas a large value of b in the 
gas equation means a large pressure (other things being equal), so 
in this case an abnormally great osmotic pressure is indicated ; 
and this carries with it (owing to a property of all isotonic solu- 
tions) an abnormal lowering of freezing-point. 

Some elementary calculus is employed in the section on chemical 
dynamics. It is of a very simple kind and does nothing except 
improve the exposition. The student is in consequence called 
upon less to accept results on trust without the argument being 
presented to him. 

Besides all the phenomena which we would expect to be dealt 


oe 


Nolices respecting New Books. 1007 


with here, an account is given of the phenomena of radioactivity. 
Curiously enough, these are given, and are the only phenomena 
described, under the head of photo-chemistry, while their photo- 
eraphic properties are barely mentioned. A short account of 
photo-chemistry would be a valuable addition and might replace 
the account of radioactivity which is not called for. 


Annuaire pour Tan 1911. Published by the Bureau des Longi- 

tudes. Paris: Gauthier-Villars. 1 fr. 50 ce. 
Bestpes the usual astronomical and geographic data, this issue 
contains several important articles. One of these, by M. A. de 
Gramont, on Stellar Spectra, replaces that of M. Cornu in previous 
issues. In it will be found a very useful summary of the present 
state of knowledge on this interesting subject. It includes a com- 
parative table of the several modes of classification adopted by 
ditferent investigators. M. H. Poincaré summarizes the contri- 
butions to the XVIth Conterence of the International Geodesic 
Association at London and Cambridge (1909); and M.G. Bigourdan 
eives particulars in regard to the total (annular) eclipses of the sun 
of 1912, the line of totality of which passes close to St. Germain 
in France. 


A Treatise on Electrical Theory and the Problem of the Unwerse. 
By G. W. vm Tunzeumann, B.Sc. Charles Griffin & Co., Ltd. 
London, 1910. 


THE appearance of this book is very timely. It brings together 
in a form fairly intelligible to a nou-mathematical reader the 
most important recent developments of modern electrical theory. 
But modern electrical theory now governs the whole domain ot 
physical science; and the time may not be far distant when 
Maxwell's joke about ‘‘the unit of life and of thought” may 
find its scientific verification in the negatively charged corpuscle. 
Some thirty or forty years ago it was a kind of accepted axiom 
that the simplicity of a theory was one of its strongest claims to 
acceptance: mais nous avons changé tout cela. The “jelly” 
theory of the ether has had to go, and ether twists have entered 
into possession. All physical laws as they appeal to our finite 
intelligences are simply statistical averages of whirling and 
drifting movements. Each so-called atom of matter is a complex 
system of discrete particles or corpuscles all in more or less rapid 
motion, acting and reacting on one another and on the mysterious 
ether in which they move. The rapidly moving negative cor- 
puscle drags along with it through the ether its trails of mag- 
netic and electric lines of force, and at every change of velocity 
starts waves of radiant energy through the ether. All this and 
a vast deal more are expounded in the pages of Mr. Tunzelmann’s 
book. Although much wider in scope and more deep seated in its 
foundations, this exposition of Electrical Theory may be compared 
to Lloyd’s treatise on the Wave Theory of Light, which in its 
day did more to spread a general knowledge of the labours of 
Young and Fresnel than any other book. So here. Mr. Tunzel- 
mann has taken a wide grasp of all the essential features of this 


a 


1008 Geoloyical Society :— 


modern theory, which by the addition of the electron to the 
electromagnetic theory as it left the hands of Maxwell has broad- 
ened in a remarkable way our whole physical horizon. The diffi- 
culties of explaining a theory which is fundamentally mathema- 
tical are not small; and althongh here and there some obscurities 
(if not inaccuracies) will be found, the general discussion is 
wonderfully clear and must have cost the author great thought 
and labour. The book proper consists of twenty-four chapters 
covering 505 pages, but a series of Appendices mainly mathema- 
tical occupies 120 pages more; and these will be found very 
useful to the student of mathematical physics. The twenty- 
fourth chapter is on the Place of Mind in the Universe, in which 
the author argues for the existence of a universal mind, corre- 
lating the otherwise separate units forming the minds of all living 
organisms, and this universal mind is regarded as the origin of 
evolution. 


Wer have received the volumes of magnetic observations during 
1905 and 1906 at Baldwin and Vieques (Porto Rico), two obser- 
vatories of the U.S. Coast and Geodetic Survey. ‘Ihe volumes 
resemble generally those for the immediately preceding years, 
containing hourly measurements of the magnetic curves, and 
diurnal inequalities based on the ten quietest days of each month. 
There are lists of the chief magnetic disturbances, and some of the 
disturbed traces are shown on a reduced scale. The Vieques volume 
also contains a list of the earthquakes recorded by Bosch-Omori 
seismographs from their erection in 1903 up to the end of 1906. 


CXV. Proceedings of Learned Societies. 
GEOLOGICAL SOCIETY. 
[Continued from p. 792.] 
March 23rd, 1910.—Prof. W. W. Watts, Sc.D., M.Sc., F.R.S., 
President, in the Chair. 
TBE following communication was read :— 


‘On Palaowyris and other Allied Fossils from the Derbyshire 
and Nottinghamshire Coalfield.’ By Lewis Moysey, B.A., M.B., F.G.S, 


April 13th, 1910.— Prof. W. W. Watts, Se.D., M.Sc., F.R.S., 
President, in the Chair. 

The following communications were read :— 

1. ‘The Volcano of Matavanu in Savaii.’ By Tempest Anderson, 
M.D., D.Sc., F.G.S. t 

Savaii is one of the German Samoan Islands in the Central Pacific 
Ocean. It is entirely volcanic, is formed of different varieties of 
basic lavas, and is for the most part fringed with coral reefs. 

The volcano of Matavanu was formed in 1905. The eruption was 
at first explosive, but since the first few wecks has been mainly 
efflusive and accompanied by the discharge of an enormous volume 
ef-very fluid basig lava, which has run by a devious course of about 


13 


a 


‘4 


, ou 


, ae 


STEGBAIIN. 


Phil. Mag. Ser. 6, Vol. 20, Pl. XIX, 


Pre. 5. 


¢ Fie. 10. 
Fria. 9. 14, 1 Fre. 11. 


Era. 12, 


re. 14. lig. 16. 
Ira. 18, 
kia. 15. Fig. 17. 


47] 
Ree 
(ance PALER Pema ee 


Phil, Mag. Ser. 6, Vol, 20, Pl. XX. 


RICHARDSON, 


Fie. 6, 


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Fria. 10. 


mee) 


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asi? 


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Fie. 8. 


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ATES, 
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Geology of the District around Llansawel. 1009 


10 miles to the sea, formed extensive fields of both slagey and 
cindery lava (pahoehoe and aa), filled up a valley to a depth in 
some places of probably 400 feet, and devastated some of the most 
fertile land in the island. The crater contains a lake, or rather 
river, of incandescent lava, so fluid that it beats in waves on the 
walls, rises in fountains of liquid basalt, and flows with the velocity 
of a cataract into a gulf or tunnel at one end of the crater. Itthen 
runs underground along a channel or channels in the new lava-field 
until it reaches the sea, into which it flows, and causes explosions 
attended with the discharge of showers of sand and fragments of 
hot lava, and the emission of vast clouds of steam. 

The many resemblances to, and few differences from, the volcano 
of Kilauea are discussed. 


Z. ‘Notes on the Geology of the District around Llansawel 


(Carmarthenshire). By Miss Helen Drew, M.A., and Miss Ida L. 
Slater, B.A. 

In this paper the authors deal with the stratigraphy and 
geological structure of a small area some 9 miles to the west of. 
Llandovery, and to the north of Llandeilo. In a brief introduction 
the reasons for the selection of this region are mentioned, and the 
work of previous observers is touched upon. 

The rocks consist of a varied series of sediments, including a 
coarse conglomerate, grits, shales, and tough blue mudstones ; 
c.eavage is almost everywhere intense. 

The beds fall naturally into three divisions, as follows :— 

( C 3. Pengelli Shales (Gala fauna). 
C. Luansawet Group. | C2. Zone of Monograptus communis. 
1. Clyn March or eyphus Grits and Shales. 
2. Llathige Shales and Mudstones. Zone of 
Mesograptus modestus. 
1. Penn-y-ddinas Grits and Shon Nicholas Con- 
: glomerates. 

A. Burtt Tew Grovr ... Beili Tew Grits and Shales. 

The stratigraphical relationships are seen most clearly in the 
highest group (C), which is therefore dealt with first. The beds 
here follow each other in perfectly regular succession, with a 
uniform strike of E. 30° N. The basal beds, with a fauna 
belonging to the zone of Monograptus cyphus, form a well-marked 
ridge across country, and Upper Birkhill and Gala Beds follow to 
the north-west. 

The second group (B) occupies a wide tract to the east of the 
Llansawel Group. The coarse basal deposits, and the characteristic. 
shales and mudstones, are described from many localities. 

Tbe lowest group (A) has its greatest development on the south 
of Llansawel. 

The structure in the eastern part of the district shows many 
points of interest, and is very much more complicated than in the 
west. The repeated outcrops of the conglomerate in the hilly 
region around Shon Nicholas are described in detail, and these give 
the clue to the structure. . 

The paper concludes with a general summary and a brief com- 
parison of this district with those of Rhayader and Pont Erwyd. 


C 
C 
te 
tae GATO GROtP . 2... 0c. B 


fr 1010 j 


INDEX to VOL. XX. 


——_—<)>_— 


AIR, on the absorption of cathode 
particles by, 520. 

Alkali sulphates, on the positive 
thermions emitted by the, 981; 
metals, on the positive thermions 
emitted by salts of the, 999. 

Allen (Dr. H. 8.) on the photo- 
electric fatigue of metals, 564. 

Alpba particles, on the number of, 
emitted by uranium and thorium, 
G91; on the probability variations 
in the distribution of, 698. 

Alternate current circuits, on a galva- 
nometer for, 509. 

Aluminium phosphate, on positive 
electrification due to beating, 575. 

Anderson (Dr. T.) on the volcano of 
Matavanu in Savaii, 1008. 

Aromatic compounds, on threefold 
emission-spectra of solid, 619, 

Atmosphere, on the amount of 
radium emanation in the, 1. 

Atoms, on the shape of, 229; on the 
mechanical vibraticn of, 657. 

Attraction constant of a molecule of 
a substance, on the, 905. 

Bakker (Dr. G.) on the thermodyna- 
mics of the capillary layer between 
the homogeneous phases of the 
liquid and the vapour, 155. 

Barkla (Prof. C. G.) on typical cases 
of ionization by X-rays, 370, 

Barton (Prof. I. H.) on the vibra- 
tion curves of violin bridge and 
strings, 456, 

Barus (C. & M.) on the interference 
of the reflected-diffiacted and 
diflracted-reflected rays of a plane 
transparent grating, 45. 

Bateman (H.) on the relation be- 
tween electromagnetism and geo- 
metry, 623; on the probability 
distribution of a particles, 704. 

Baynes (R. E.) on Mr. Bateman’s 
paper on earthquake-waves, 664. 

Beatty (R. T.) on the production of 
cathode particles by homogeneous 
Roéntgen radiations and their ab- 
sorption by hydrogen and air, 320. 

Berry (G. H.) cn the pianoforte 
sounding-board, 662. 


Bessel functions of imaginary prgu- 
ment, on the approximate caleula- 
tion of, 988, 

Beta particles, on the absorption and 
reflexion of the, by matter, 879; 
on the reflexions of homogeneous, 
of different velocities, 866, 

— rays, on the range of, 880; on 
the homogeneity of the, fiom 
radium I, 870, 

Blakesley (‘T. HH.) on a means of 
measuring the apparent diameter 
of the pupil of the eye in yery 
feeble light, 966. 

Boilivg-point, on relations between 
the physical properties of liquids 
ut the, 522. 

Books, new :—Tissot’s Les Oscilla- 
tions électriques, 247; Jonguet’s 
Théorie des Moteurs Thernnques, 
247; Bulletin of the Bureau of 
Standards, 882; Kelvin’s Mathe- 
matical and Physical Papers, vol. 
iv., 540; Crabtree’s Klementary 
Treatment of the Theory of Spin- 
ning ‘Tops and Gyroscope Motion, 
542; Jahnke and Emde’s Funk- 
tionentafeln mit Formeln und Kur- 
ven, 542; Jones’s Intreduction to 
Physical Chemistry, 1006; An- 
nuaire pour l’an 1911, 1007; de 
Tunzelmann’s Treatise on Elec- 
trical Theory and the Problem of 
the Universe, 1007; U.S. Coast 
and Geodetic Survey, 1008, 

Bosworth (1. O.) on metamorphism 
around the Ress of Mull granite, 
790. 

Boyle (Dr. R. W.) on the volatiliza- 
tion of radium emanation at low 
temperatures, 95+. 

Bragg (Prof. W. IL) en the cor- 
puscular hypothesis of the y and 
X rays and the range of f rays, 
385. 

Brass, on the specific charge of the 
ions emitted by hot, 556. 

Canalstrahlen, on the, 742. 

Capillary layer between homogeneous 
phases of liquid and vapour, on 
the, 136. 


| 
. 


INDEX. 


Carslaw (Prof. H.8.) on the scatter- 
ing of waves by a cone, 690. 

Cathode particles, on the production 
of, by ontgen radiations and 
their absorption by hydrogen and 
air, 320. 

Charcoal, on the absorption of radium 
emanation by, 778. 

Chattock (Prof. A. P.) on the forces 
at the surface of a needle-point 
discharging in air, 266; on the 
loulzing processes at a voint dis- 
charging in air, 277. 

Cheneveau (C.) on the magnetic 
balance of MM. P. Curie and 
C. Cheneveau, 357. 

_Chree (Dr. C.) on results obtained 
at Kew Observatory with an 
Elster and Geitel electrical dissipa- 
tion apparatus, 475. 

Circuits, on musical are oscillations 
in coupled, 660. 

Coker (Prof. E.G.) on the optical 
determination of stress, 740. 

Concave surfaces, on the clinging of 
sound-waves to, 1001. 

Cone, on the scattering of waves 
by a, 690. 

Continuity of the liquid and gaseous 
states of matter, on the equation 
of, 665. 

Convection of heat from a body 
cooled by a stream of fluid, on 
the, 591. 

Cook (G.) on a_ hydrodynamical 
illustration of the theory of the 
transmission of aerial and electrical 
waves by a grating, 303. 

Cooke (Prof. H. L.) on the heat 
developed during the absorption 
of electrons by platinum, 173. 

Copper, on the specific charge of the 
ions emitted by hot, 548.- 

Currents, on the study of variable, 
by means of the phaseograph, 969. 

Cylindrical lenses, on the lengths of 
the focal lines of, 59. 

Dielectric sphere, on the accelerated 
motion of a, 828. 

Diffraction grating, ou a method of 
counting the rulings of a,714; on 
the groove-form and energy dis- 
tribution of, 886. 

Dixon (E. E. L.) on the carboni- 
ferous succession in Gower, 791, 
Donaldson (H.) on the problem of 
uniform rotation treated on the 

principle of relativity, 92. 


1OLL 


Doublet, on the motion of a particle 
about a, 380. 

Drew (Miss H.) on geology of dis- 
trict around Llansawel, 1009. 

Earthquake-waves, on, 664. 

Ebblewhite (T’. F.) on the vibration 
curves of violin bridge and strings, 
456. 

Kecles (Dr. W. H.) on an oscillation 
detector actuated by resistance- 
temperature variations, 125; on 
the energy relations of certain 
detectors used in wireless tele- 
oraphy, 533. 

Echelette grating for the infra-red, 
on the, 470; on infra-red investi- 
gations with the, 898. 

Elastic sphere, on the effect of radial 
forces in opposing the distortion 
of an, 437. 

Electric current required to fuse a 
wire, on the, 607. 

currents, on the study of 

variable, by means of the phaseo- 

eraph, 969. 

discharge of a point in air, on 

the, 266, 277. 

dissipation apparatus, on results 
obtained at Kew Observatory with 
an Elster and Geitel, 475. 

—- doublet, on the motion of an 
electrified particle near an, 544. 

organ, on the eye as an, 560. 

waves, on the bending of, round 
a large sphere, 157 ; on the trans- 
mission of, by a erating, 303. 

—— wind, on the reaction of the, 276. 

Electricity, on rays of positive, 782 ; 
on the, of mercury- falls, 903. 

Electrification due to heating alu- 
minium phosphate, 573. 

Electrified sphere, on the accelerated 
motion of an, 610. 

Electromagnetism and geometry, on 
the relation between, 629. 

Electromotive force, on the Weston 
cell as a standard of, 206. 

Electron, on the motion of a, about 
a doublet, 380. 

—— orbits, on the analysis of the 
radiation from, 642. 

theory of the optical properties 
of metals, on the, 835. 

Electronic, on molecular and, poten- 
tial enerey, 249. 

Electrons, on the heat developed 
during the absorption of, by plati- 
num, 175, 


_MOe Fe 


1012 INDEX. 


Elster and Geitel electrical di-sipa- 
tion apparatus, on results obtained 
at Kew Observatory with an, 475. 

Emission-spectra of aromatic com- 
pounds, on the, 619. 

Energy, on the flow of, in an inter- 
ference field, 299), 

Ether, on the critical phenomena of, 
793. 

Ewing (Prof. J. A.) on magnetic 
hysteresis, 1005. 

Evans (E. J.) on the deflexion by a 
magnetic field of radium B on 
recoil from radium 4, 882. 

Eye, on the, as an electrical organ, 
60; ona means of measuriny the 
apparent diameter of the pupil of 
the, in very feeble light, 966, 

Fields of force, on the intensity of 
periodic, 844, 

Finlayson (A. M.) on ore-deposition 
in the lead and zine veins of Great 
Britain, 543, 

Fletcher (A. L.) on the radioactivity 
of the rocks of the Transandine 
tunnel, 36. 

Florance (D. C. H.) on primary and 
secondary y rays, 9z1. 

Galvanometer for alternate current 
circuits, on a, 309. 

Gamma rays, on the homogeneity of 
the, 248, 383 ; on the corpuscular 
hypothesis of the, 385; on primary 
and secondary, 921. 

Garrett (A. E.) on positive electri- 
fication due to heating aluminium 
phosphate, 573. 

Gas thermometer, on a_ constant 
pressure, 296. 

Gaseous, on the equation of con- 
tinuity of the liquid and, states of 
matter, 665. 

Geiger (Dr. H.) on the number of 
a particles eniitted by uranium and 
thorium and by uranium minerals, 
691; on the probability variations 
in the distribution of a particles, 
698. 

Geological Society, proceedings of 
the, 548, 790, 1007. 

Geometry, on the relation between 
electromagnetism and, 623. 

Gibson (Prof. A. H.) on a formula 
for the discharge over a_broad- 
crested weir, 95, 


_ Gold, on the specific charge of the 


ions emitted by hot, 550, 


Goldstein (Prof. E.) on three‘old 
emission-spectra of solid aromatic 
compounds, 619, 

Grating, on the interference of ray 
from a plane transparent, 45; a 
hydrodynamical illustration of the 
transmission of aerial and electri- 
cal waves by a, 803; on a method 


of counting the rulings of a dif- 


fraction, 714; on the echelette, 
for the infra-red, 770; on groove- 
form and energy distribution of 
diffraction, 886; on infra-red jn- 
vestigations with the echelette, 898, 

Gray (J. A.) on the heterogeneity 
of the B rays from a thick layer of 
radium Jf, 870, 

Greenhill (Sir G.) on pendulum 
motion and spherical trigono- 
metry, 728. 

Groove-form and energy distribution 
of diffraction-gratings, on, 886. 

Harrison (W. J.) on the stability of 
superposed streams of viscous 
liquids, 493. 

ITeat, on the convection of, from a 
body cooled by a stream of fluid, 
591, ; 

Hot bodies, on the specific charge of 
the ions emitted by, 545. 

Houstoun (Dr, R.A.) on the damping 
of long waves in a rectangular 
trough, 247. 

Hulbirt (E. R.) on the specific charge 
of the ions emitted by hot bodies, 
545, 

Hydrogen, on the absorption of 
cathode particles by, 320, 

Hysteresis loops and Lissajous’ 
figures, on, 417, 

Infra-red, on investigations in the, 
with the echelette grating, 770, 
898. 

Interference of rays from a plane 
transparent grating, on the, 45. 
field, on the flow of energy in 

an, 290. 

Interferometer, on an, 45, 

Ionization, on typical cases of, by 
X-rays, 370; on the, produced by 
the splashing of mercury, 464, 903. 

Ionizing processes at a point dis- 
charging in air, on the, 277. 

Tons, on the specific charge of the, 
emitted by hot bodies, 545, 

Iron, on the specific charge of the 
ions emitted by hot, 552. 


PNG DSECX 


Jeans (Prof. J.) on the motion of a 
particle about a doublet, 380; on 
the analysis of the radiation from 
electron orbits, 642; on non- 
Newtonian mechanical systemsand 
Flanck’s theory of radiation, 945. 

Jolley (A. C.) on the magnetic 
balance of MM. P. Curie and 
C. Cheneveau, 366. 

Joly (Prof. J.) on the amount of 
thorium in sedimentary rocks, 126, 
353. 

Jones (Prof. E. T.) on musical are 
oscillations in coupled circuits, 660. 

Kave (Dr. G. W.C.) on a method 
of counting tie rulings of a dif- 
fraction grating, 714; on the ex- 
pansion and thermal hysteresis of 
fused silica, 718. 

Kleeman (Dr. R. D.) on the shape 
of the atom, 229; on the homo- 
geneity of the y rays of radium, 
248; on the shape of the mole- 
cule, 445; on the equation of 
continuity of the liquid and gaseous 
states vf matter, 665; on mole- 
cular attraction, 901; on the 
attraction constant of a molecule 
of a substance aud its chemical 
properties, 905. 

Kovarik (Dr. A.) on the absorption 
and reflexion of ®8-particles by 
matter, 849; on the reflexion of 
homogeneous B-particles of ° dif- 
ferent velocities, 866. 

Lagging of pipes and wires, on tlie, 
oll. 

Lamb (Prof. H.) on a hydrodynami- 
cal illustration of the theory of 
the transmivsion of aerial and 
electrical waves by a grating, 303. 

Larmor (Sir J.) on the. statistical 
theory of radiation, 350. 

Lees (Dr. C. H.) cn the laws of the 
direction of thermo-electrie cur- 
rents, 384. 

Lenard (Dr. P.) on the electricity of 
mercury falls and very large ions, 
903. 

Lenses, on the lengths of the focal 
lines of cylindrical, 59; on a for- 
mula for the spherical aberration 
in, 82. 

Lewis (Dr. W. C. McC.’ on the 
nature of the transition layer be- 
tween two adjacent phases, 502. 

Lewis (Prof. W. J.) ou wiltshireite, 

' anew mineral, 474, 


1013 


Light, ou a difference in the photo- 
electric effect caused by incideut 
and emergent, 531. 

Liquid mixtures, on partial pressures 
in, 97; and gaseous states of 
matter, on the equation of con- 
tinuity of the, 665. 

Liquids, on the stability of super- 
posed streams of viscous, 493: on 
the molecular pressure in, 502; 
on relations between physical pro- 
perties of,at the boiling-point, 522. 

Lissajous’ figures, on hysteresis loops 
and, 417. 

Lonsdale (J. J.) on the ionization 
preduced by the splashing of mer- 
cury, 464. 

Love (Prof. A. Ix. H1.) on the effest 
of radial forces in opposing the 
distortion of an elastic sphere, 
437, 

Magnetic balance of MM. P. Curie 
and C. Cheneveau, on the, 357. 
field, on the electrustatic etect 
of a changing, 384; on the de- 
flexion by a,of radium B on recoil 

trom radium A, 882. 

hysteresis, on, 1005, 

Makower (Dr. W.) on the deflexion 
by an electrostatic field of radium 
B on recoil from radium <A, 875; 
on the deflexion by the magnetic 
field of radium B on recoil from 
radium A, 882. 

Martin (EH. R.) on the lagging of 
steam-pipes, 018. 

Mason (Prof. M.) on the flow of 
energy ian interference field, 290. 

Mechanical systems, on non-New- 
tonian, and Planck’s theory of 
radiation, 948. 

Mercury, on the ionization produced 
by the splashing of, 464, 908 ; 
on the series spectrum of, 636. 

Metals, on the specific charge of the 
ions emitted by hot, 545; on the 
photoelectric fatigue of, 564; on 
the electron theory of the optical 
properties of, 835. 

Miller (Dr. W.) on a constant pres- 
sure gas- thermometer, 296. 

Mills (Dr. J. E.) on molecular attrac- 
tion, 629. 

Milner (Dr. S. R.) on the series 
spectrum of mercury, 636. 

Mineral, on wiltshireite, a new, 474, 

Minerals, on the ratio between ura- 
nium and radium in, 345. 


P tide Se 


1014 EN DEX: 


Molecular and electronic potential 
energy, on, 249; attraction, on, 
629, 901. 

Molecule, on the shape of the, 445; 
on the attraction constant of a, of 
a substance, 908. 

Musical arc oscillations in coupled 
circuits, on, G60. 

Nicholson (Dr. J. W.) on the bend- 
ing of electric waves round a large 
sphere, 157; on the accelerated 
motion of an electrified sphere, 
610; on the accelerated motion of 
a dielectric sphere, 828; on the 
approximate calculation of Bessel 
functions of imaginary argument, 
938. 

Nichrome, on the specific charge of 
the ions emitted by hot, 557. 

Nickel, on the specific charge of the 
ions emitted by hot. 548. 


‘Non-Newtonian mechanical systems 


and Planck’s theory of radiation, 
943 

Optical determination of stress, on 
the, 740; properties of metals, on 
the electron theory of the, 835. 

Oscillation detectors, actuated by 
resistance-temperature variations, 
on, 128: on the energy relations 
of certain, used in wireless tele- 
eraphy, 635. 

Oscillations in coupled circuits, on 
musical are, 660, 

Osmium, on the specific charge of 
the ions emitted by hot, 549. 

Palladium, on the specific charge of 
the ions emitted by hot, 548. 

Pendulum motion and spherical tri- 
gonometry, on, 728. 

Phaseograph, on the study of vari- 
able currents by means of the, 969. 

Phillips (W. C.'S.) on a galvano- 
meter foralternate current ¢ circuits, 
309. 

Photoelectric effect, on a difference 
in the, caused by incident and 
emergent hglt, 331. 

fatigue of metals, on the, 564. 

Physical properties of liquids at the 
boiling-point, on relations between 
the, 522. 

Pianoforte sounding-board, on the, 
652. 

Pipes, on the lagging of, 51]. 

Pirret (Miss R.) on the ratio between 
uranium and radium in minerals, 
345, 


Planck's theory of radiation, on non- 
Newtonian mechanical systems 
and, 943. 

ty latinum, on the heat developed 
during ‘the absorption of electrons 
by, 173; on the specific charge of 
the ions emitted by hot, 546. 

Point discharging in air, on the 
forces at the surface of a, 266; on 
the ionizing processes at a, 277. 

Porter (Prof. “A. W. ) on the lagging 
of pipes and wires, 511, 

Positive electricity, on rays of, 752. 

electrification due to heating 
aluminium phosphate, on, 573. 

Potential energy, on molecular and 
electronic, 249, 

Prescott (J.) on the effect of radial 
forces in opposing the distortion 
of an elastic sphere, 437. 

Pressures, on partial, im liquid mix- 
tures, 97, 

Pupil of eye, on a means ot measur- 
ing the apparent diameter of the, 
in very feeble light, 966, 

Radiant emission from the spark, on 
a new, 707. 

Radiation, on the theory of, 121, 238, 
350, 904; on the mechanical pres- 
sure of, 538; on the analysis oo 
the, from electron orbits, 642; 0 
non-Newtonian mechanical Be 
tems and Planck’s theory of, 
943. 

Radioactivity of the rocks of the 
Transandine tunnel, on the, 36, 
‘adium, on the homogeneity of the 
y rays of, 248, 383; on the rela- 
tion between uranium and, 340; 
on the ratio between uranium and, 
in minerals, 845; on the hetero- 
geneity of the 8 rays from a thick 
layer of Kal, 870; on the de- 
flexion by an electrostatic field of 
Ra B on recoil from Ra A, 875; 
on the deflexion by a magnetic 
field of Ra B on recoil from Ra A, 

882. 


emanation, on the amount of, © 


in the atmosphere, 1: on the ab-— 


; 


sorption of, by coconut charcoal, 
778; on the volatilization of, ag 
low ‘temperatures, 955. 


Rayleigh (Lord) on the finite vibra * 


h 


tions of a system about a configu-— 


ration of equilibrium, 450; on the 
problem of the whispering gallery, 
1001. 


DN-DE XxX. 1015 


Refraction by non - homogeneous 
media, experiments on, 712. 

Relativity, on the problem of uni- 
form rotation treated on the 
principle of, 92. 

Richardson (Prof. O. W.) on the 
heat developed during the absorp- 
tion of electrons by platinum, 173 ; 
on the specific charge of the ions 
emitted by hot bodies, 545; on the 
positive thermions emitted by the 
aikali sulphates, 981; on the posi- 
tive thermions emitted by the salts 
of the alkali metals, 999. 

Roberts (D. EK.) on musical are oscil- 
lations in coupled circuits, 660. 
Rocks, on the radioactivity of the, 
of the Transandine tunnel, 36; on 
the amount of thorium in sedi- 

mentary, 125, 353. 

Rountgen radiations, on the pro- 
duction of cathode particles by, 
320. 

Rotation, on the problem of uniform, 
treated on the principle of relati- 
vity, 92. 

Russ (Dr. 8.) on the deflexion by an 
electrostatic field of radium B on 
recoil from radium A, 875. 

Russell (Dr. A.) on the convection 
of heat froma body cooled by a 
stream of fluid, 591. 

Rutherford (Prof. E.) on the number 
of a particles emitted by uranium 
and thorium and by uranium 
minerals, 691; on the probability 
variations in the distribution of 
a particles, 698. 

Satterly (J.) onthe amount of radium 
emanation in the 1:ower regions of 
the atmosphere, 1; on the absorp- 
tion of radium emanation by coco- 
nut charcoal, 778. é 

Siegbahn (M.) on the study of vari- 
able currents by means of the 
phaseograph, 969. 

Silica, on the expansion and thermal 
hysteresis of fused, 718. 

Silver, on the specific charge of the 
ions emitted by hot, 548. 

Slater (Miss I. L.) on geology of 
district around Llansawel, 1005. 
Smart (EK. H.) on a formula for the 
spherical aberration in a lens- 

system, 82. 

Smith (Dr. 8. W. J.) on the Weston 
cell as a standard of electromotive 
force, 206. 


Soddy (F.) on the relation between 
uranium and radium, 340; on the 
rays and product of uranium X, 
342; on the ratio between ura- 
nium and radium in minerals, 345; 
on the homogeneity of the y rays 
of radium, 383. 

Sounding-board, on the pianoforte, 
652. 

Sound-waves, on the clinging of, to 
concave surfaces, 1001. 

Spark, on a new radiant emission 
from the, 707. 

Spectra, on the emission-, of aro- 
matic compounds, 619. 

Spectrometer, on a vacuum, 768. 

Spectrum, on the series, of mercury, 
636; on displacements in the, due 
to pressure, 788. 

Sphere, on the bending of electric 
waves round a large, 157; on the 
effect of radial forces in opposing 
the distortion of an elastic, 437, 
445; on the accelerated motion of 
an electrified, 610; on the accele- 
rated motion of a dielectric, 828. 

Spherical aberration in a lens-system, 
on a formula for the, 826. 

—- trigonometry, on pendulum 
motion and, 728. 

Stead (G.) on the problem of uniform 
rotation treated on the principle 
of relativity, 92. 

Steam-pipes, on the lageing of, 518. 

Steel, on the specific charge of the 
ions emitted by hot, 556. 

Stephenson (A.) on displacements in 
the spectrum due to pressure, 788 ; 
on the intensity of periodic fields 
of force, 844. 

Story (Prof. W. Ii.) on partial pres- 
sures in liquid mixtures, 97. 

Stress, on the optical determination 
of, 740. 

Stuhlmann (O., jr.) on a difference 
in photoelectric effect caused by 
incident and emergent light, 331. 

Sulphates of the alkali metals, on the 
positive thermions emitted by the 
931. 

Sumpner (Dr. W. I.) on a galvano- 
meter for alternate current cir- 
cuits, 309. 

Surface forces, on the theory of, 135. 

Sutherland (W.) on molecular and 
electronic potential energy, 249 ; 
on the mechanical vibration of 
atoms, 657, 


fe err 


LE Oe 


1016 


Tantalum, on the specific charge of 
the ions emitted by hot, 554. 

Telegraphy, on the energy relations 

‘of certain detectors used in, 533. 

Thermions, on the positive, emitted 
by the alkali sulphates, 981; on 
the positive, emitted by the salts 
of the alkali metals, 399. 

Thermometer, on a constant pressure, 
296. 

Thompson (Prof. 8. P.) on hysteresis 
loops and Lissajous’ figures, 417. 
Thomson (Sir J. J.) on the theory 
of radiation, 238; on the motion 
of an electrified particle near an 
electrical doublet, 544; on rays of 

positive electricity, 752. 

Thorium, onthe amount of, in sedi- 
mentary rocks 125, 353; on the 
number of a particles emitted by, 
691. 

Thornton (Prof. W. M.) on the eye 
as an electrical organ, 560, 

Trensition layer,on the nature of the, 

between two adjacent phases, 5!)2. 

Tiigonometry, on pendulum motion 
and spherical, 728. 

Trowbridge (Prof. A.) on a vacuum 
spectrometer, 768; on the groove- 
furm and energy distribution of 
diffraction gratings, 886; on infra- 
red investigations with the eche- 
lette grating, 898. 

Tungsten, on the specific charge of 
the ions emitted by hot, 555. 

Tunzelmann (G. W. de) on the me- 
chanical pressure of radiation, 538. 

Tyndall (A. M.) on the ionizing 
processes at a point discharging in 
air, 277. 

Tyrer (D.) on relations between the 
physical properties of liquids at 
the boiling-point, 522. 

Uranium, on the relation between, 
and radium, 340; on the ratio be- 
tween, and radium in minerals, 
345; on the number of a particles 
emitted by, 691. 

Uranium X, on the rays and product 
of, 342. 

Vaughan (A.) on the carboniferous 
succession in Gower, 791. 

Vibration curves of violin bridge and 
strings, on the, 456. 

——-, on the mechanical, of atoms, 
657. 


EN DEX, 


Vibrations, on the finite, of a system 
about a configuration of equili- 
brium, 450. 

Violin bridge and strings, on the 
vibration curves of, 456. 

Viscous liquids, on the stability of 
superposed streams of, 493. 

Waves, on the transmission of aerial 
and electrical, by a. grating, 303 ; 
on earthquake, 664; on the scat- 
tering of, by a cone, 690; on the 
clinging of sound-, to concave 
surfaces, 1001. 

Weir, on a formula for the discharge 
over a broad-crested, 95. 

Weston cell as astandard of electro- 
motive force, on the, 206, 

Whispering gallery, on the prublem 
of the, 1V01. 

Whitehead (Prof, J. B.) on the 
electrostatic effect of a changing 
magnetic field, 384. , 

Whitwell (A.) on the lengths of the 
focal lines of cylindrical lenses, 
59, 


Wilson (Prof. H. A.) on the statis- 


tical theory of heat radiation, 121, 
904; on the electron theory of the 
optical properties of metals, 835, 
Wilson (W.) on the reflexion of 
homogeneous 8-particles of differ- 
ent velocities, 866; on the hetero- 


geneity of the 8 rays from a thiek 


layer of radium EK, 870, 

Wiltshireite, on the new mineral, 
474, 

Wireless telegraphy, on the energy 
relations of certain detectors used 
in, 533. 

Wires, on the lagging of, 51]. 

Wood (Prof. R. W.) on a new 
radiant emission from the spark, 
707; experiments on refraction by 
non-homogeneous media, 712; on 


the echelette grating for the infra- 


red, 770 ; on the groove-form and 
energy distribution of diffraction 
gratings, 886; on infra-red in- 
vestigations with the echelette 
grating, 898. ; 

X-rays, ou typical cases of ioniza- 
‘tion by, 370; on the corpuscular 


hypothe-is of, 385, 


Young (EF. B.) on the critical phe- 


nomena of ether, 793, 


END OF THE TWENTIETH VOLUME.  - 


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